TOPIC AREA: NUMBERS AND OPERATIONS

UNIT 1: Mathematical operations on whole numbers up to 100 000

1) Place value of digits
1) What is the place value pf each digit in the number 147,306,429,185?
Solution:
Millions Thousands Units

H T O H T O H T O

2 4 7 3 0 6 4 2 9

Ones
Tens
Hundreds
Thousands
Ten thousands
Hundred thousands
Millions
Ten millions
Hundred millions
2) What is the place value of 3 in the number 243,109,764?
243,109,764
Millions
3) What is the rank of 4 in the number 2,468,903,127?
2,468,903,127
Hundred millions
4) Write the position of 6 in the number 46,219.
46,219
Thousands

Note: Place value = Rank = Position

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 2) Value of digits.
To get the value of any digit, multiply it by its place value.
Examples.
1) Find the value of each digit in the number 273,109,864,529.
Solution:
Number Place value Value

1 0 9, 8 6 4, 5 2 9
Ones 7×1=7
Tens 2 × 10 = 20
Hundreds 5 × 100 = 500
Thousands 4 × 1, 000 = 4,000
Ten thousands 6 × 10,000 = 60,000
Hundred thousands 8 × 100,000 = 800,000
Millions 9 × 1,000,000 = 1,000,000
Ten millions 0 × 10,000,000 = 0
Hundred millions 1 × 100,000,000= 200,000,000

2) What is the value of 4 in 346,290,127?

346,290,127
4 × 10,000,000 = 40,000,000
3) Find the sum of the values of 3 and 5 in the number 23,659,186.
23,659,186
Value of 5 = 5 × 10,000 = 50,000
Value of 3 = 3 × 1,000,000 = 3,000,000
Sum = 3,000,000 + 50,000 = 3,050,000
4) Calculate the difference between the value of 2 and 8 in the number 48,917,253.
Value of 2 = 2 × 100 = 200
Value of 8 = 8 × 1,000,000 = 8,000,000
Difference = 8,000,000 – 200 = 7,999,800
5) Find the quotient of values of 6 and 3 in the number 68,236,197.
Value of 6= 6 × 10,000,000 = 60,000,000
Value of 8 = 3 × 10,000 = 30,000
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 Sum = 60,000,000 ÷ 30,000 = 2,000

3) Reading and writing numbers in words.

 4,670,200 = Four million, six hundred seventy thousand two hundred.
 700,070,007 = seven hundred million, seventy thousand, seven.
 109,637,390 = one hundred nine million, six hundred thirty seven thousand,
three hundred ninety.
 3,802,716 = three million, eight hundred two thousand seven hundred sixteen.

4) Writing numbers in figures.

Write the following numbers in figures:
1) Sixty billion, six hundred million, six thousand, sixty.
Billions Millions Thousands Units

H T O H T O H T O H T O

6 0 6 0 0 0 0 6 0 6 0

60,600,006,060
2) Seven million, seventy thousand, seven hundred.
7,070,700
Exercises
1) What is the position of 7 in the number 37,089,231?

2) Give the place values of the underlined digits.

a) 646, 290,317

b) 231,038,478

c) 56,251,029

3) Find the value of 6 in the number 346,098,124.

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4) What are the place value and the value of 7 in the number 276,109,841?

5) What are the place value of 2 and the value of 9 in the number 269,208,465?

6) Write the following numbers in words:

a) 2,009

b) 8,000,0008

c) 11,101,111

d) 84,678,239

7) Write the numbers below in figures.

a) One thousand, nine hundred fifty five
b) Four hundred thirty million, six hundred forty.
c) Twenty million, four thousand, nine hundred.
8) Find the sum of the values of 5 and 7 in the number 254,720,189.
9) Calculate the difference between the values of 6 and 4 in the number 24,068,915.
10) What is the product of the values of 3 and 9 in the number 23,597?
11) Find the quotient of the values of 9 and 3 in the number 436,859,102,847.

5) EXPANDED FORM
a) Expanding in place value form.
Expand the numbers in place value form.
1) 245,304 =(2 × 100,000) + (4 × 10,000) + (5 × 1,000) + (3 × 100) + (0 × 10) + (4 × 1)

2) 350,841,256 = (3×100,000,000) + (5×10,000,000) + (0×1,000,000) + (8×100,000) +

(4×10,000) + (1×1,000) + (2×100) + (5×10) + (6×1)

b) Expanding in multiples of 10.

Expand the numbers below in multiples of 10.
1) 324,689 = (3 ×100,000) + (2×10,000) + (4×1,000) + (6×100) + (8×10) + (9×1)
= (3×10×10×10×10×10) + (2×10×10×10×10) + (4×10×10×10) +

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 (6×10 ×10) + (8×10) + (9×10)
2) 48,901 = (4×10,000) + (8×1,000) + (9×100) + (0×10) + (1×1)
= (4×10×10×10×10) + (8×10×10×10) + (9×10 ×10) + (0×10) + (1×10)

c) Expanding in value form.

Expand these numbers in value form.
1) 2,678,045 = (2×1,000,000) + (6×100,000) + (7×10,000) + (8×1,000) + (0×100) +
(4×10) + (5×1)
= 2,0000,000 + 600,000 + 70,000 + 8,000 + 0 + 40 + 5

3) 456,109,867 = (4×100,000,000) + (5×10,000,000) + (6×1,000,000) + (1×100,000) +

(0×10,000) + (9×1,000) + (8×100) + (6×10) + (7×1)
= 400,000,000 + 50,000,000 + 6,000,000 + 100,000 + 0 +9,000 + 800 +
60 + 7
d) Expanding in power form.
Expand the following numbers in power form.
1) 2,430,798 = (2×1,000,000) + (4×100,000) + (3×10,000) + (0×1,000) + (7×100) +
(9×10) + (8×1)
= (2×106) + (4×105) + (3×104) + (0×103) + (2×102) + (9×101) + (8×100)

2) 425,769 = (4×100,000) + (2×10,000) + (5×1,000) + (7×100) +

(6×10) + (9×1)
= (4×105) + (2×104) + (5×103) + (7×102) + (6×101) + (9×100)

5) Finding the expanded number

1) Which number has been expanded to give the following?
(4×105) + (2×104) + (5×103) + (7×102) + (6×101) + (9×100)
400,000
20,000
+ 5,000
700
60
9
425,769

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2) Find the number that was expanded to become 500,000 + 40,000 + 2,000 + 600 + 0 +
8
500,000
40,000
+ 2,000
600
0
8
542,608

EXERCISES
1) Expand the numbers below in value form.
a) 43,908
b) 234,618
c) 4,089,356
2) Expand the following numbers in power form.
a) 340,728
b) 97,129,935
c) 240,143,176
3) Peter added 543,829 and 245,108. Expand the answer he got I power form.
4) The population of Rwanda is 13,278,914. Expand the number is value form.
5) Which number has been expanded to give (2×105) + (8×104) + (5×103) + (0×102) + (9×
101) + (1×100)?
6) Find the number that has been expanded to give 3,000,000 + 200,000 + 20,000 +
5,000 + 800 + 40 + 3.
7) Felix expanded a number and got (9×106) + (5×105) + (2×104) + (1×103) + (8×102) + (5×
101) + (7×100)

6) Forming numerals from digits

In Hindu Arabic numerals system there are 10 digits such as 1, 2, 3, 4, 5, 6, 7, 8, 9 and 0.
All numbers are formed using these above digits.

Examples
1) Write down all numbers that can be formed from the digits 2, 7 and 5
275 725 527

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 257 752 572
2) Write down all numbers that can be formed using the digits 6, 4, 0 and 8
6408 4068 8046
6480 4086 8064
6048 4806 8604
6084 4860 8640
6804 4608 8406
6840 4680 8460
3) What is the smallest number that can be formed from the digits 8, 6, 0, 9 and 2?
Smallest number= 20,689
4) Give the largest number that can be formed from the digits 5, 1, 7, 0, 6 and 4
Largest number = 765,410
5) Find the sum of the lowest number and the biggest number that can be formed from
the digits 4, 8, 0, 3, 6 and 7.
Smallest number= 30,478
Largest number = 87,430
Sum = 30,478 + 87,430 =117,908

Exercises
1) Form two six digit numbers from the digits: 1, 2, 3, 4, 5 and 6.
2) Give the lowest seven digits number formed from 4, 3, 0, 8, 5, 1, 9.
3) Find the difference between the biggest number and the smallest number that can be
formed using the digits 6, 2, 9, 0, 4 and 7.
4) What is the sum of the lowest number and the largest number formed from the digits
8, 1, 5, 0, 7, 3 and 9.
5) Write two even numbers that can be formed from the digits 7, 4 and 3.
6) Form the largest and the smallest number using the digits 6, 8, 2, 7 and 5.
7) Give any three odd numbers that can be formed from the digits 8, 3, 9 and 2.
8) Write down all numbers that can be formed using 4, 8 and 2.

8) Complements of numbers
A complementary number is a number that can be added to another to make a rounded
figure.
A rounded figure is number that is equal to 10; 100; 1,000; 100,000; 1,000,000 etc
 The complementary number to 4 is 6 because 4 + 6 = 10
 The complement of 27 is 73 because 27 + 73 = 100

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 The complement of 516 is 484 because 516 + 484 = 1000

Examples
Find the complements of the following numbers;
a) 68 = 100 – 68 = 32
b) 451 = 1000 – 451 = 549
c) 2,891 = 10,000 – 2,891 = 7,109
d) 72 = 100 – 72 = 28
e) 4 = 10 – 4 = 6

Exercises
1) Find the complements of the following numbers;
a) 47
b) 382
c) 19
d) 7
e) 7,148
f) 28,495

2) What is the complementary number of 467?

3) Find the complement of 1,748.
4) What should be added to 34,617 to give 50,000?
5) Robert had 482 Frw and wanted to buy a dozen of pens at 600 Frw. How much money
does he need?
6) Agape planned to plant 10,000 seedlings. He only has 7,218 seedlings. How many
more seedling does he need?
7) Calculate the complementary number to 57.
9) Rounding off whole numbers
 If the figure on the right of the required place value is less than 5 (0,1,2,3,4), add 0 to
the required place value.
 If the figure on the right of the required place value is 5 or greater than 5 (5,6,7,8,9)
add 1 to the required place value.
 All digits behind the required place value become zeros.

Examples

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1) Round off 268 to the nearest tens.
268
+1__
270
2) Round off 3,749 to the nearest hundred.
3,749
+0__
3,700
3) Round off 43,524 to the nearest thousand.
43,524
+1____
44,000
4) Correct 173,897 to the nearest ten thousands.
173,897
+1____
170,000
Exercises
1) Round off 39,621 to the nearest thousands.
2) Manzi bought a radio at 25,875 Frw. Round the amound of money to the nearest ten
thousand.
3) Find on number that when rounded to the nearest hundreds is 453 000.
4) A number rounded to the nearest hundreds is 6,700.
a) Determine the lowest possible number.
b) Determine the highest possible number.
5) The average number of pupils in primary schools is 3,489,989. Round this number to
the nearest thousand.
6) Agatesi bought her car for 9,561,000 Frw. Round off the money she paid to the
nearest million.
7) Add 24,896 and 73,586 and round the result to the nearest thousand.
10) Comparing whole numbers using ¿ ,>¿ or =
When comparing we use:
 ¿ : greater than
 ¿ : less than
 ¿ : equal to
To compare two or more numbers, first, count the number of digits in each number.
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1) If the given numbers have different number of digits, the one having more digits is
greater.
Examples
Compare the following numbers.
a) 46,798 ¿ 123,012
b) 23,401 ¿ 9,876
c) 238,012,741 ¿ 89,376,874

2) If the given numbers have the same number of digits, compare digit by digit from
the left, until the two digits in the same corresponding place value tell which
number is greater.
Examples
Compare the numbers below.
d) 274,692 ¿ 274,892
a) 372,817,092 ¿ 372,817,091
b) 845,127 = 845,127

Exercises
1) Write true or false.
a) 127,398 is less than 98,753
b) 381,645 is greater than 381,641
c) A number with more digits is greater than another number with less digits.
2) Fill in the spaces with the correct symbol of comparison.
a) 610 487 7 248 160
b) 713 482 917 713 482 907
c) 541 845 541 845
3) Mutesi has 468 379 Frw and John has 98 476 Frw. Who has less money?
4) Which number is greater, 595 06 or 607 899? Explain.
5) Camille harvested 5,562 tonnes of beans and Felix harvested 5,256 tonnes of beans.
Who harvested more beans?
6) Gasabo district collected 45,853,925 Frw in taxes while Gakenke collected 9,756,895
Frw. Which district collected less money?

11) Ordering numbers

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Numbers are arranged in two ways;
1) Ascending /increasing order: means to arrange numbers from the smallest to the
largest.
2) Descending/decreasing: means to arrange numbers from the largest to the
smallest.

Examples
1) Arrange the following numbers in ascending order.
468 274 ; 468 374 ; 466 274 ; 468 284
Solution:
466 274; 468 274; 468 284; 468 374

2) Re-arrange the numbers below I decreasing order.

25 649; 25 639; 25 648; 256 481; 25 688
Solution:
256 481; 25 688; 25 649; 25 639

Exercises
1) Underline the smallest number.
33 333; 3 333; 333 333

2) Arrange the following numbers in ascending order.

a) 457 782; 457 792; 459 682; 98 765

b) 1,673,421; 1,065,345; 1,671,241; 1,065,234

c) 750,236,172; 750,237,172; 750,236,072; 750,236,174

3) Bank notes are numbered in order from A2408700 to A2408719.

a) How many notes are there altogether?
b) If each note is 5,000 Frw, how much money is in the bundle?

4) Order from the highest to the lowest.

a) 45,238; 110,210; 45,338; 45,239

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 b) 12,042,994; 12,420,994; 12,994,609; 12,499,906

12. OPERATIONS IN WHOLE NUMBERS

1) Addition
Addition basically means putting together.
When adding two or more numbers, first arrange digits in the numbers according to their
place values (ones under ones, tens under tens, …….). Always start adding from ones and
carry where necessary to the larger place value.
Examples
1) Add 346,897 and 49,736
346,897 36 Adding
+ 49,736 Note: + 12 Adding
396,736 48 ∑ ¿ ¿

2) A factory made 1,358,916 nails on Monday and 963,078 nails on Tuesday. Find the
total numbers of nails which were made by the factory in two days.
1,358,916
+963,078
2,321,994

Exercises
1) Peter bought a radio at 434 890 Frw, a suit at 34 878 Frw and a book at 7 396 Frw.
How much money did he spend altogether?
2) Betty deposited money in Umurenge SACCO as follows: 720,654 F in January,
1,004,529 F I February and 3,894,728 F in March.
a) Calculate the total deposit Ingabire made in the Umurenge SACCO.
b) Why is it necessary to deposit money in the bank?
3) What is the sum of 38,298,784 and 25,734,928?
4) Increase 5,702,854 by 4,589,627.
5) 4,836 and 1,689 make what?
6) A poultry farmer sold 252 797 chickens in one year. The next year he sold 391 358
chickens. The third year he sold 198 524 chickens. How many chickens did he sell in
three years?

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7) There were 246 240 books in a library and 167 645 more books were donated to the
same library. How many books are there altogether?
8) Complete:
. 8 .9 467. 1.34
a) + 3 6 9 . b) + 2. 4 6 c) + .7 .6
6.42 .5.1 763.

d) 68 + . = 152 e) . + 728 = 1,420

Properties of addition
1) Commutative property
The result after adding numbers in any order remains the same.
Examples
Add a) 20 + 40 =
Skill 1: 20 + 40 = 60
Skill 2: 40 + 20 = 60
Therefore 20 + 40 = 40 + 20
b) 70 + 30 = 100
Skill 1: 70 + 30 = 100
Skill 2: 30 + 70 = 100
Therefore 70 + 30 = 30 + 70
The same sum is got by adding in either order
A+B=B+A

2) Associative property
Addition of problems involving more than two numbers, any two numbers added first do
not change the result.
Examples
Add a) 40 + 30 + 60
Skill 1: (40 + 30) + 60 = 70 + 60 = 130
Skill 2: (60 + 40) + 30 = 100 + 30 = 130
Skill 3: (30 + 60) + 40 = 90 + 40 = 130

We can arrange three or more addends in any order and still get the same sum
(A + B) + C = (A + C) + A = (B + C) + A
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 Examples
1) Fill in the missing numbers
a) 7 + 4 = +7
b) 4 + (5 + ) = (6 + 4) + 5
2) Find the value of a
a) (a + 20 ) + 10 = (10 + 20) + 6
b) 40 + (15 + a) = 15 + (40 + 10)

2) Subtraction
Subtraction means to take away a number from another.
Examples
1) Subtract 8,456,782 from 10,200,420
10,200,420 36 Minuend
−8,456,782 Notice: −12 Subtrahend
1,743,638 24 Difference

2) Thomas’ salary was 127,400 Frw. It was reduced by 49,680 Frw. How much does he
get now?
127,400 Frw
−49,680 Frw
77,720 Frw
Exercises
1) What is the difference between 624 415 and 35 897?
2) Subtract the following:
a) 6,000,101 – 4,999,011 =
b) 3,642,110 kg – 1,563,276 kg =
c) 8,621,143 trees from 9,132,423 trees =
3) Ingabire had a debt of 7,683,942 Frw. If she pays 5,839,678 Frw, how much debt is
left?
4) The population of a country grew from 6 784 512 to 9 201 076 in two years. What
was the population increase over this period of time?
5) 3,567,342 babies were born in a country in 2016. Of these, 1,593,599 babies were
girls. Find the number of boys.
6) A truck carrying 2,560,000 litres of milk was in an accident. 1,756,950 litres were
split. How much milk remained?

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7) A farmer harvested 8,320,165 kg of maize. By the end of the month, he had sold
6,826,759 kg. How much kilograms are still in his store?
8) Subtract 34,763 from 82,124.
9) By how much is 367 015 greater than 346 929?
10) What should be added to 7 248 974 to get 8 124 610?
11) Complete:
534 . 8 2 .6 7.32
a) −2 .7 9 b) −. 4 8 . c) −26 . 4
.6 .7 4.59 .75.

d) 84 – . = 28 e) . – 352 = 119
3) Multiplication
Multiplication is a process of adding a number to itself a particular number of times.
Examples: If we say 10 × 6 we mean 10 + 10 + 10 + 10 + 10 + 10 until we get six times.
Examples
1) Multiply: 25,467 × 289
25,467
×289
24 Multiplicand
229203
+ 203736
Notice: ×3 Multiplier
24 Product
50934
7359963

Casting out 9
56,738
× 472
113476
5+6+7+3+8=2+9=1+1=2
2) Multiply and prove your answer:
2 56,738 × 472 + 397166
8 88 4+7+2=1+3=4
226952
4 2+6+7+8+0+3+3+6=3+5=8
26780336
4×2=8

Exercises
1) When multiplying 408 by 23, Peter forgot to use 0 of 408. Find the total error he made.
2) A train carries 1 640 passengers a trip. How many passengers will it carry if it makes 15
trips?
3) One book cost 427 Frw. How much do 378 similar books cost?

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4) A parade of soldiers was made up of 87 rows. There are 50 soldiers in each row. How
many soldiers were there?
5) There are 24 bottles in a crate of Soda. How many bottles are there in 75 crates of
Soda?
6) How many eggs are there in 59 trays of eggs?
7) How many books are there in 54 dozens of books?
8) A man multiplied 728 by 36 and forgot to use 8 of 728. Calculate the total error he
made.
9) What is the product of 639 and 87?
10) Complete:
a) 35 × . = 840
b) . × 15 = 390
Properties of multiplication.
1) Commutative property
The result after multiplying two numbers in any order remains the same.
Examples: a) Multiply: 20 × 8
Skill 1: 20 × 8 = 160
Skill 2: 8 × 20 = 160
Therefore 20 × 8 = 8 × 20
b) Multiply: 15 × 4
Skill 1: 15 × 4 = 60
Skill 2: 4 × 15 = 60
The same Therefore
product is 15got
× 4 = 4 × 15
by multiplying two numbers in either order
A ×B = B × A
2) Associative property
Multiplication problems involving more than two numbers, any two numbers multiplied
first, the result remains the same.
Example: Multiply: 4 × 8 × 5
Skill 1: (4 × 8) × 5 = 32 × 5 = 160
Skill 2: (5 × 4) × 8 = 20 × 8 = 160
Skill 3: (8 × 5) × 4 = 40 × 4 = 160

To multiply three or more numbers, the first two numbers multiplied first, do not
change the result. (A×B) ×C = (A ×C) × A = (B × C) × A

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 3) Distributive property
To work out numbers using distributive property, distribute multiplication with numbers
inside the brackets using addition and subtraction only
Examples: a) Calculate: 4(5 + 6)
4(5 + 6) = 4 x 11 = 44
b) Work out: 3 × 8 – 3 × 2
3 × 8 – 3 × 2 = 3(8 – 2) = 3 × 6 = 18
c) Calculate: 25 × 64 + 25× 36
25 × 64 + 25× 36 = 25(64 + 36)
=25 × 100 = 2 500
d) Evaluate: 65 × 1548 – 65 × 548
65 × 1548 – 65 × 548 = 65(1548 – 548)
= 65 x 1000 = 65,000
Exercise
1) Fill in the missing numbers
a) 39x(82+ )=39x100
b) 76 × (163 – ) = 7,600
2) Work out using distributive property.
a) 82 × 726 + 82 × 274

b) 1836 × 48 – 836 × 48

Quick multiplication
1. Quick multiplication by 10, 100, 1000, etc
To multiply any number by 10, 100, 1000, 10 000, 100 000 etc, simply add the zeros to
the given number.
Examples
Calculate:
a) 25 x 10 = 250
b) 386 x 1000 = 386,000
c) 74 x 100 = 7,400
d) 5 x 10 000 = 50 000

2. Quick multiplication by 2
To multiply a number by 2, add the same number to itself.
Examples
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Work out:
a) 243 x 2 =
243
+ 243
486
b) 2579 x 2
2579
+ 2579
5158

3) Quick multiplication by 5
To multiply a number by 5, multiply it by 10 and divide the result by 2
Examples
Effectuate: a) 275 × 5
275 × 5 = (275 × 10)÷ 2
= 27500 ÷ 2 = 1 380
b) 789 × 5
789 × 5 = (789 × 10) ÷ 2
= 7890 ÷2
=3 945

4. Quick multiplication by 50
To multiply a number by 50, multiply it by 100 and divide the result by 2.
Examples
Work out:
a) 437 ×50
437 ×50 = (437 × 100) ÷ 2
= 43700 ÷2
=21 850
a) 689 ×50
689 ×50 = (689 × 100) ÷ 2
= 68900 ÷2
=34 450
5. Quick multiplication by 25
To multiply a number by 25, multiply it by 100 and divide the result by 4.
Examples
Work out: a) 746 ×25
746 ×25 = (746 × 100) ÷4
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 = 74600 ÷4
=18 650
b) 379 ×25
379 ×25 = (379 × 100) ÷4
= 37900 ÷4
=9 475
6. Quick multiplication by 20
To multiply a number by 20, multiply it by 2 and then by 10.
Examples
Calculate: a) 475 × 20
475 × 20 = 475 × 2 × 10 = 950 × 10
=9 500
b) 247×20
247×20 = 247×2 ×10
= 494 × 10 = 4 940
7. Quick multiplication by 9
To multiply a number by 9, multiply it by 10 and subtract the original number from the
result.
Examples
Work out: a) 354 × 9
354 × 9 = 354 × (10 – 1)
=354 × 10 – 354 × 1
=3540 – 354
=3 186
b) 672 × 9
672 × 9 = 672 × (10 – 1)
=672 × 10 – 672 × 1
=6720 – 672
=6 048
8. Quick multiplication by 99
To multiply a number by 99, multiply it by 100 and subtract the original number from the
result.
Examples
Work out: a) 396 × 99
396 × 99 = 396× (100 – 1)
=396 × 100 – 396 × 1
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 =39600 – 396
=39 204
b) 2 485 × 99
2 485× 99 = 2 485 × (100 – 1)
=2 485 × 100 – 2 485 × 1
=2 48500 – 2 485
=246 015
9. Quick multiplication by 19
To multiply a number by 19, multiply it by 20 and subtract the original number from the
result.
Examples
Calculate: a) 846 × 19
846 × 19 = 846 × (20 – 1)
= 846 × 20 – 846 × 1
= 16 920 – 646
= 16 074
b) 758 × 19
758 × 19 = 758 × (20 – 1)
= 758 × 20 – 758 × 1
= 15 160 – 758
= 14 402
10. Quick multiplication by 49
To multiply a number by 19, multiply it by 50 and subtract the original number from the
result.
Examples
Calculate: a) 837 × 49
837 × 49 = 837 × (50 – 1)
= 837 × 50 – 837 × 1
= 41 850 – 837
= 41 013
b) 528 × 49
528 × 49 = 528× (50 – 1)
= 528 × 50 – 528 × 1
= 26 400 – 528
= 25 872

11. Quick multiplication by 11

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To multiply a number by 11, multiply it by 10 and add the original number to the result.
Examples
Calculate: a) 368 × 11
368 × 11 = 368 × (10 + 1)
= 368 × 10 + 368 × 1
= 3680 + 368
=4 048
b) 5 849 × 11
5 849 × 11 = 5 849 × (10 + 1)
= 5 849 × 10 + 5 849 × 1
= 58 490 + 5 849
= 64 339
4) Division
1. Division without a remainder
Division is a process of finding out how many times one number is contained in another.
Examples
1) Divide: 19 248 ÷ 8 = 02 406 003 –138
02 406 0 103
8 19248 30 –92
–0 30 115
19 00 – 115
–16 000
32 3) Divide: 52 728 ÷ 13 = 4 056
–32 4 056 9 Quotient
004 13 52 728
3 27 Dividend
–0 –52
48 007
48 –0 Divisor
00 72 5) Calculate: 1382010÷ 35
2) Divide 208 830 by 6 –65 39 486
034605 78 35 12 82010
6 208 830 – 78 –0
–0 00 138
20 –105
–18 4) Work out: 60 835 ÷ 23 332
28 2645 –315
–24 23 60835 170
048 –46 – 140
– 48 148 301

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 280 4 208 188
210 235 988 880 –0
210 –940 1880
00 488 – 1880
6) Divide 988 880 by 235 –470 0
Exercises 4) 54 964 ÷ 28 =
Divide the numbers: 5) 12 700 314 ÷ 27 =
1) 8 744 480 ÷ 215 =
6) 88 831÷211 =
2) 3008488 ÷ 124 =
7) 162 828 ÷ 36 =
3) 4 575 244 ÷ 68 =
8) 33 088 120 ÷ 95 =
2. Division with a remainder
1) Divide 6 425 628 ÷ 24 918
267734 –852
24 6425628 66 Remainder
–48
162
–144
185
–168
176
– 168
082
72
108
96
12 Remainder

2) Divide 3 486 728 by 213

16374
213 3486728
–213
1356
–1278
787
–629
1582
– 1491

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 Word problems involving division
1) When X is divided by 62, the quotient is 12 and the remainder is 9. Find the value of X.
X = dp + r
= (62 × 12) + 9
= 744 + 9
= 953
2) The government released 36 450 000 Frw to 50 youth societies. How much did each
get? Total = 36 450 000 Frw
Number of societies = 50
Each got = 36 450 000 Frw ÷ 50 = 729, 000 Frw

Exercises
1) A total of 54 142 books were distributed to 23 classes. How many books did each class
get?
2) Share 2,026,800 Frw among 24 employees.
3) A soda bottling company packed 8,462,376 bottles of soda in crates each containing
24 bottles. Find the number of crates that were packed.
4) A sugar factory manufactured 12,960,648 kg of sugar in a year. How many kg of sugar
were produced every month if the factory produces equal amounts of sugar monthly?
5) The electricity board put the lamp posts at a distance of 484 dm apart in the
distribution of power to all roads in the city. How many posts were put along a street
covering a length of 5 246 560 dm?
Divisibility test for numbers
Divisibility test refers to the shortest possible process through which to determine
whether or not a given number can be divided by another without a remainder.
1) Divisibility test for 2
A number I divisible by 2 if it ends with an even number.
Examples
1) Is 46 734 divisible by 2?
Yes, it is divisible by 2 because the last digit is an even number
2) Among the following numbers which ones are divisible by 3?
67 094 , 22 229 , 400 001 and 17 930
Solution: 67 094 and 17 930 are divisible by 2

2) Divisibility test for 3

A number is divisible by 3 if the sum of its digits is divisible by 3.
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 Examples
1) Find out whether 5,241 is divisible by 3.
Sum of digits = 5 + 2 + 4 + 1 = 12
It is divisible by 3
2) Is 57,068 divisible by 3? Show how you reach the answer.
Sum of digits = 5 + 7 + 0 + 6 + 8 = 26
It is not divisible by 3

3) Divisibility test for 6

A number is divisible by 6 if it is divisible by both 2 and 3.
Examples
1) Find out if 93,546 is divisible by 6.
 It is divisible by 2 because it ends with an even number.
 Sum of digits = 9 + 3 + 5 + 4 + 6 = 27
It is divisible by 3
 Therefore, it is divisible by 6
2) Without using a long calculation show if 70,314 is divisible by 6.
 It is divisible by 2 because it ends with an even number.
 Sum of digits = 7 + 0 + 3 +1 + 4 = 15
It is divisible by 3
 Therefore, it is divisible by 6

4) Divisibility test for 4

A number is divisible by 4 if the last two digits are divisible by 4.
Examples
1) Without dividing show whether or not 295,672 is divisible by 4.
The last two digits are 72 and 72÷ 4 = 18
It is divisible by 4
2) Among the numbers below which ones are not divisible by 4?
44 449 ; 71 536 , 80 802 and 53 952
Solution: 44 449 and 80 802 are not divisible by 4.

5) Divisibility test for 12

A number is divisible by 12 if it is divisible by both 3 and 4.
Examples
1) Find out if 11,148 is divisible by 12.
 Sum of digits: 1 + 1 + 1 + 4 + 8 = 15

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 It is divisible by 3
 The last two digits are 48 and 48÷ 4 = 12
It is divisible by 4
 Therefore, it is divisible by 12
2) Check whether 315,936 is divisible by 12.
 Sum of digits: 3 + 1 + 5 + 9 + 3 + 6 = 27
It is divisible by 3
 The last two digits are 36 and 36÷ 4 = 9
It is divisible by 4
 Therefore, it is divisible by 12

6) Divisibility test for 5

A number is divisible by 5 if it ends with 0 or 5.
Examples
Among the following numbers, which ones are divisible by 5?
55 558 ; 719 420 ; 400 004 ; 39 175
Solution: 719 420 and 39 175 are divisible by 5

7) Divisibility test for 8

A number is divisible by 8 if the last three digits are divisible by 8.
Examples
From the numbers below, choose those that are nit divisible by 8.
8 047 288 ; 715 000 ; 48 480 024 and 88 884
Solution: 8 047 288 and 88 884 are not divisible by 8.

8) Divisibility test for 9

A number is divisible by 9 if the sum of its digits is 9.
Examples
1) Is 12,222 divisible by 9? Show why.
Sum of digits = 1 + 2 + 2 + 2 + 2 = 9
It is divisible by 9
2) By not dividing, show if 84,753 is divisible by 9.
Sum of digits = 8 + 4 + 7 + 5 + 3 = 27
It is divisible by 9

9) Divisibility test for 10

A number is divisible by 10 if it ends with 0.
Examples
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Circle the numbers that are divisible by 10.
; 1 010 101 ; and 80 005
576 930 79 360

10) Divisibility test for 11

A number is divisible by 11 if the difference between the sum of odd position numbers
and even position numbers is 0 or divisible by 11.
Examples
1) Is 383,482 divisible by 11?
3*83*,48*2
 Sum 1 = 3 + 3 + 8 = 14
 Sum 2 = 8 + 4 + 2 = 14
 Difference = 14 – 14 = 0
 It is divisible by 11
2) Find out if 928 092 divisible by 11.
9*28*,09*2
 Sum 1 = 9 + 8 + 9 = 26
 Sum 2 = 2 + 0 + 2 = 4
 Difference = 26 – 4 = 22
 It is divisible by 11

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UNIT 2: POSITIVE AND NEGATIVE INTEGERS

INTEGERS

Definition: Integers are positive and negative numbers together with zero plotted in
equal distances on number line.

A. Integers on a number line

Study the number line below.

-7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7
A set of integers include:
1) Positive integers (1, 2, 3, 4, 5, 6, ………….)
2) Zero (0)
3) Negative numbers (-1, -2, -3, -4, -5, -6, ………)
Notice: a) Zero is neither negative nor positive integer.
b) Any integer without a sign is a positive integer.
→ +6 is the same as 6 as 10 is the same as +10.
B. Explaining integers
 If I have no money at all, it means 0 money.
 If I have been given some 100 F, it means +100 F.
 If I have lost 2,000 F, it means -2,000 F.
 If a team scores 2 goals, it means +2.
 If a trader makes a profit of 5,000 F, it means +5,000 F.
 If a shopkeeper makes a loss of 3,000 F, it means -3,000 F.
C. Distance between two integers
i) Two integers on the same side of zero
If two integers are on the same side of zero, the distance between them is the difference
of their magnitude.
Magnitude of a number, is the distance from 0 to that number.
Study the number below and state the magnitude of: a) -3
b) +4
3 steps 4 steps

-7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7
Solution:
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a) The magnitude of -3 = 3
b) The magnitude of +4 = 4

Examples
1) What is the distance between + and +10?
 Magnitude of +2 = 2
 Magnitude of +0 = 10
 Difference=10 – 2 = 8 steps
2) Find the distance between -6 and -13.
 Magnitude of -6 = 6
 Magnitude of -13 = 13

ii) Two integers on opposite sides of zero

If two numbers are on opposite sides of zero, the distance between them is the sum of
their magnitudes.
Examples
1) What is the distance between -5 and +4?
 Magnitude of -6 = 6
 Magnitude of +4 = 4
 Sum = 6 + 4 = 10 steps
2) Find the distance between -10 and +10.
 Magnitude of -10 = 10
 Magnitude of +10 = 10
 Sum = 10 + 10 = 20
D. Arrows on the number line

Note: * If the arrow looks at the left, the answer is a negative.

* If the arrow looks at the right, the answer is a positive.
* To find the length of the arrow, just find the distance between two numbers.

Examples
1) What integers are represented by the arrow?
f
a b

-7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7
d c
e
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Solution
 Magnitude of -2 = 2
 Magnitude of -6 = 6
 Difference = 6 – 4 = 2
Arrow a indicates -3

 Magnitude of 0 = 0
 Magnitude of +4 = 4
 Difference = 4 – 0 = 4
Arrow b indicates +4

 Magnitude of +1 = 1
 Magnitude of +6 = 6
 Difference = 6 – 1 = 5
Arrow c indicates -5

 Magnitude of -1 = 1
 Magnitude of -4 = 4
 Difference = 4 – 1 = 3
Arrow d indicates =+3

 Magnitude of -5 = 5
 Magnitude of +4 = 4
 Sum = 4 + 5 = 9
Arrow e indicates +9

 Magnitude of -4 = 4
 Magnitude of +3 = 3
 Sum = 4 + 3 = 7
Arrow f indicates -7

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2) How far is it from -9 to -7?
 Magnitude of -9 = 9
 Magnitude of -7 = 7
 Difference = 9 – 2 = 2
It is +2
E. Comparing of integers
Study the number line below.

-7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7

NOTE:
1) On a number line, the integers are always in order from the smallest to the
biggest.
2) Any integer on the right is greater than any integer on its left on a number line.
3) Any integer on the left is smaller than any integer on its right on a number line.
4) All positive integers are greater than all negative integers.
5) Zero “0” is greater than all negative integers.

Examples.
Compare the following integers:
a) + 2 ¿ -9
b) -100 ¿ +6
c) +80 ¿ -5,000
d) -10 ¿ -30
e) -80 ¿ 0
f) 1 ¿ -9

F. Ordering integers

Descending order (decreasing order) means to arrange from the biggest number to the
smallest number.
Ascending order (increasing order) means to arrange from the smallest number to the
biggest number.
Examples
1) Arrange the following numbers in descending order.
-6 , +1 , -12 , -7 , +3 , -15 , 0 , -4
Answer: +3 , +1 , 0 , -4 , -6 , -7 , -12 , -15

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2) Re-arrange the numbers below in increasing order.
7 , -5 , -10 , 0, -3 , -9 , 6
Answer: -10 , -9 , -5 , -3 , 0 , 6 , 7

G. Opposites or inverses
Study the number line below.

-7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7

 The opposite of -1 = +1
 The opposite of -2 = +2
 The opposite of +3 = -3
 The opposite of -4 = +4
 The opposite of +5 = -5
 The opposite of -6 = +6
 The opposite of +7 = -7

NOTE: When two numbers have the same magnitudes, and the two numbers are on
opposites of zero, one number is the opposite of another.

Additive inverses
Any integer added to its opposite the result is zero.
Examples
1) +3 + -3 = 0
2) +7 - 7 = 0
3) (-13) + (+13) = 0
4) -8 + 8 =0
Then, * +3 is an additive inverse of -3
* -30 is an additive inverse of +30
* +100 is an additive inverse of -100
* -80 is an additive inverse of +80
G. Operations of integers

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 1. Addition and subtraction
 When adding or subtracting two integers that have the same signs, simply add
them and keep their sign.
Examples
Work out: +6 +3 = +9
–4 – 2 = –6
+8 + 2 = +10
–5 – 6 = –11
 When adding or subtracting two integers that have different signs, simply subtract
them and write the sign of a big number.
Examples
Work out: +5 – 2 = +3
–6 + 3 = –3
+9 –14 = – 5
–3 + 7 = +4
Adding and subtracting integers using a number line
Work out the following using a number line.
1) +3 + 2 = +5 +3 +2

-7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7
= +5
2) +2 – 4 = –2
+4
+2

-7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7
=–2
3) –3 + 5 = +2
+5
–2

-7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7
=+2

More about addition and subtraction.

Work out the following:
a) –2 + 4 – 6 = –2 – 6 + 4
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 =–8+4
= –4

b) (–4) + (+7) – (+10) + (+5) + (–12) = –4 + 7 –10 + 5 – 12

= –4 – 10 – 12 + 7 + 5
= –26 + 12
= –14

c) (+3) – (+6) + (–4) – (–7) + (+4) + (–3) + (+9) = +3 – 6 – 4 + 7 + 4 – 3 + 9

=+3+7+9–6–4–3
= + 19 – 13
=+6

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UNIT 3: CLASSIFYING NUMBERS BY THEIR PROPERTIES
1. Whole numbers
Whole numbers are numbers with no fraction and begins with zero. Whole numbers tell
us how many members a group has.
Eg: {0, 1, 2, 3, 4, 5, 6, 7, …………}

2. Counting numbers
Counting numbers are numbers used while counting the number of members a group
has.
Eg: {1, 2, 3, 4, 5,…………….}
Notice: Zero is not a counting number.

3. Even numbers
Even numbers refer to the numbers which are divisible by 2.
Eg: {0, 2, 4, 6, 8, 10, 12,……………….}

4. Odd numbers
Odd numbers refer to the numbers which are not divisible by 2.
Eg: {1, 3, 5, 7, 9, 11, 13, 15, 17, 19,………………….}

5. Ordinal numbers
Ordinal numbers are numbers that tell us the positions of members of a group
Eg: 1st, 2nd, 3rd, 4th, 5th, ……..

6. Consecutive numbers
Consecutive numbers refer to numbers which follow one another in series.

There are four types of consecutive numbers:

i) Consecutive counting numbers eg: 1, 2, 3, 4, 5, 6, 7, 8,……………..
ii) Consecutive even numbers eg: 0, 2, 4, 6, 8, 10,………………………
iii) Consecutive odd numbers eg: 1, 3, 5, 7, 9, 11, ……………………..
iv) Consecutive prime numbers eg: 2, 3, 5, 7, 11, 13, 17, 19, ………….

7. Square numbers
Square numbers are numbers got by multiplying a number by itself.
Eg: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 225, …………………..
a) Finding the square of numbers
To get the square of any number, multiply it by itself.

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 Examples
 The square of 1 is 12 = 1 x 1 = 1
 The square of 2 is 22 = 2 x 2 = 4
 The square of 3 is 33 = 3 x 3 = 9
 The square of 12 is 122 = 12 x 12 = 144
 The square of 20 is 202 = 20 x 20 = 400
 The square of 150 is 1502 = 150 x 150 = 22 500

b) Square of numbers ending in 5

To get the square of numbers ending in 5, write immediately 25 at the end and multiply
the remaining number in front of 5 by the number that follows if in whole numbers.
Examples
Find the square of the following numbers:
a) Square of 15 = 225
1x2=2
b) Square of 45 = 20 25
4 x 5 = 20
c) Square of 75 = 5625
7 x 8 = 56
d) The square of 10 = 11025
10 x 11 = 110

c) Square root
 The square root of any number is a number which was multiplied by itself to get
that number.
 The symbol of square root is √
Examples
1) Calculate the square root of 64
2 64
2 32
2 16 Then √ 64 = 2 x 2 x 2 = 8
2 8
2 4
2 2
1

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2) Find the number that was multiplied by itself to get 144.
2 144
2 72
2 36 √ 144 = 2 x 2 x 3 = 12
2 18
3 9
3 3
1

Exercises
1) Find the square root of 625
2) Find the number that is multiplied by itself to be 81
3) Calculate: √
169+ √ 225
2
√100+ √256
4) Work out:
√4
√289−√ 121
5) Find:
√ 25

8. Prime numbers
Prime numbers refer to numbers which have only two factors, one and itself.
Eg: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, ……………

Notice: *The smallest prime number is 2

*The only one prime number which is even number is 2
Exercises
1) What is the sum of the prime numbers between o and 20?
2) Find the sum of the prime numbers between 20 and 30
3) Calculate the average of the prime numbers between 20 and 40.
4) Find the mean of the prime numbers between 10 and 20

TOPIC 5: PRIME FACTORISATION

Prime factorisation is a way of finding the prime factors of a number.

When prime factorising:

1) We only use prime numbers
2) We always start from the lowest prime number to the largest

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 Examples
1) Prime factorise 96
2 96
2 48
2 24
2 12
2 6
3 3
1

2) Express 252 as a product of its prime

factors
2 252
2 126
3 63
3 21
7 7
1
252 = 2 × 2 × 3 × 3 × 7

Exercises
1) Express 216 as a product of its prime
factors.
2) Prime factorise 336.
3) Express 2520 as a product of its
prime factors.
4) Express 420 as a product of its prime
factors.
5) Prime factorise 1980 and write it as a
product of its prime factors.
6) Express 910 as a product of its prime
factors.

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 Finding the prime factorised number
1) Find the number which has been prime factorised to get 2 x 2 x 2 x 3 x 5
2 x 2 x 2 x 3 x 5 = 120
The number is 120
2) Which number is factorised to get 22 x 32 x 5?
2 2 x 32 x 5 = 2 x 2 x 3 x 3 x 5
= 180
The number is 180
3) Find the number whose factorization s 5 x 5 x 7 x 17
5 x 5 x 7 x 17 = 2 975
The number is 2 975

Finding the unknown prime factor

1) The prime factors of 60 are 2 x 2 x p x 5. Find the value of p.
2 60 2x2xpx5
2 30
3 15 or p = 60 ÷ ( 2 x 2 x 5)
3 5 = 60 ÷ 20
1 =3

Therefore, p = 3
2) The prime factors of 90 are 2 x 3 x 3 x n. Find the value of n.
n = 90 ÷ (2 x 3 x 3 )
= 90 ÷ 18
=5

MULTIPLES AND FACTORS OF NUMBERS

1) Multiples of numbers
*A multiple is a number that contains another number the exact number of times.
*To get the multiples of any number, multiply it by counting numbers.
Examples
 The multiples of 2 are = {2, 4, 6, 8, 10, 12, 14, 16, 18, ………..}
 The multiples of 5 are = {5, 10, 15, 20, 25, 30, 35, ……………….}
 The multiples of 8 are = {8, 16, 24, 32, 40, 48, 56, ………………}
 The multiples of 12 are = {12, 24, 36, 48, 60, 72, ……………….}

a) More about multiples

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1) Write down the multiples of 5 between 20 and 40.
={25, 30, 35}
2) List down the multiples of 6 less than 40.
={6, 12, 18, 24, 30, 36}
3) Write the first six multiples of 7.
={7, 14, 21, 28, 35, 42}
4) How many multiples of 9 are there between 20 and 60?
={27, 36, 45, 54}
There are 4 multiples
5) What is the sum of multiples of 5 between 10 and 30?
={15, 20, 25}
Sum = 15 + 20 + 25
= 60
6) Find the sum of the multiples of 2 less than 10.
={2, 4, 6, 8}
Sum = 2 + 4 + 6 + 8
= 20
b) Common multiples
1) List the common multiples of 4 and 6 less than 50.
 Multiple of 4 = {4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48}
 Multiples of 6 = {6, 12, 18, 30, 36, 42, 48}
 Common factors = {12, 24, 36, 48}
2) Find the common multiples of 8 and 12 less than 80
 Multiples of 8 = {8, 16, 24, 32, 40, 48, 56, 64, 72}
 Multiples of 12 = {12, 24, 36, 48, 60, 72}
 Common multiples = {24, 48, 72}
3) What is the Lowest Common Multiple of 10 and 15?
 Multiples of 10 = { 10, 20, 30, 40, 50, 60, 70, 80, …….}
 Multiples of 15 ={15, 30, 45, 60, 75, 90, ………}
 Lowest Common Multiple is 30
4) Find the LCM of 20 and 30.
 Multiples of 20 = { 20, 40, 60, 80, …….}
 Multiples of 30 ={30, 60, 90, 120, ……..}
 Lowest Common Multiple is 60

2) Factors or Divisors
A factor or a divisor is a number that divides another exactly (without a remainder).

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Notice: * 1 is a factor of every number.
* Any number is a factor of itself.
Examples
1) List down all factors of 12
1 x 12
2x6
3x4
Factors of 12 ={1, 2, 3, 4, 6, 12}
2) What are the divisors of 20?
1 x 20
2 x 10
4x5
Factors of 12 ={1, 2, 4, 10, 20}
3) Write down all factors of 50.
1 x 50
2 x 25
5 x 10
Factors of 12 ={1, 2, 5, 10, 25, 50}
4) How many factors does 40 have?
1 x 40
2 x 20
4 x 10
5x8
Factors of 12 ={1, 2, 4, 5, 8, 10, 20, 40}
It has 8 factors
a) More about factors
1) What is the sum of factors of 8?
Factors of 8 ={1, 2, 4, 8}
Sum =1 + 2 + 4 + 8
=15
2) Find the sum of the factors of 15
Factors of 15 ={1, 3, 5, 15}
Sum =1 + 3 + 5 + 15
=24
b) Common factors or divisors
1) Find the common divisors of 16 and 20.
 Factors of 16 ={1, 2, 4, 8, 16}
 Factors of 20 ={1, 2, 4, 5, 10, 20}
 Common factors ={1, 2, 4}
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2) What are the factors of 24 and 36?
 Factors of 24 ={1, 2, 3, 4, 6, 8, 12, 24}
 Factors of 36 ={1, 2, 3, 4, 6, 9, 12, 18, 36}
 Common factors ={1, 2, 3, 4, 6, 12}
3) What is the greatest common factor of 30 and 45?
 Factors of 30 ={1, 2, 3, 5, 6, 10, 15, 30}
 Factors of 45 ={1, 3, 5, 9, 15, 45}
 Greatest Common Factors is 15
4) Find the Greatest Common Factor of 40 and 60.
 Factors of 40 ={1, 2, 4, 5, 10, 20, 40}
 Factors of 60 ={1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60}
 Greatest Common Factors is 20

3) Lowest Common Multiple and Greatest Common factor

1) Find the LCM and GCF of 20 and 30
2* 20 30
2 10 15
3 5 15
5* 5 5
1 1
LCM = 2 x 2 x 3 x 5 = 60
GCF = 2 x 5 = 10
2) What is the LCM and GCF of 60, 80 and 120?
2* 60 80 120
2* 30 40 60
2 15 20 30
2 15 10 15
3 15 5 15
5* 5 5 5
1 1 1
LCM = 2 x 2 x 2 x 2 x 3 x 5 = 240
GCF = 2 x 2 x 5 = 20

UNIT 4: FRACTIONS
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 1. The notion of a fraction
Definition: A fraction is a part of the whole.
Study the following and state the fractions representing the shaded part.

2 4 1
=6 = 10 =3

2. The parts of a fraction

A fraction has three main parts: - Numerator
- Denominator
- Fraction bar
numerator
4
eg: 9 fraction bar
denominator
3. Types of a fraction
There are three types of fractions: - Proper fractions
- Improper fractions
- Mixed fractions
a) Proper fractions
A proper fraction is a fraction whose numerator is smaller than the denominator.
1 7 5 3
eg: 5 ; 20 ; 9 ; 14 ; ………
b) Improper fractions
An improper fraction is a fraction is whose numerator is bigger than the denominator.
4 9 12 15
eg: 3 ; 7 ; 11 ; 4 ; ………
c) Mixed fractions
A mixed fraction is a fraction made up of a whole number and a fraction.
1 3 5
eg: 1 2 ; 2 5 ; 4 11 ; ………
4. Writing and writing fractions
When reading a fraction, the numerator is read as a cardinal number while the
denominator is read as an ordinal number.
Write the following fractions in words:
2 1
a) 5 = two fifths d) 10 = a tenth
3 5
b) 8 = three eights e) 12 = five twelves
7 7
c) 9 = seven ninths f) 20 = seven twentieths
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 11 1
g) 50 = eleven fiftieths h) 3 = a third

1 3
Exception: * 2 = a half * 4 = three quarters
1 2
* 4 = a quarter * 4 = two quarters
5. Changing improper fractions to mixed fractions
4
1) Change 3 into a mixed fraction.
4 1
3
= 1
3
9
2) Express 5 as a mixed fraction.
9 4
5
= 1
5

6. Changing mixed fractions into improper fractions

3
1) Change 1 5 into an improper fraction.
3 8
1
5 5
=
1
2) Express 4 6 as a mixed fraction.
1 25
4 =
6 6
7. Expressing fractions as decimals
Express the following fractions as decimals:
2
a) 5 = 0.4
1
b) 2 = 0.25
3
c) 4 = 0.75
5
d) 8 = 0.625
1
e) 2 5 = 2.2

8. Changing decimals into fractions

Express the following decimals as fractions in its lowest terms.
4
a) 0.4 = 10
2
=5

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 35
b) 3.5 = 10
7 1
= 2 = 32
53
c) 0.053 = 1000
75
d) 0.75 = 100
3
=4
225
e) 2.25 = 100
1
=2 4
9. Comparing fractions
 When two fractions have the same numerators, the one having a small denominator is
a greater one.
3 3
eg: * 5 ¿ 4

8 8
* 15 ¿ 30

6 6
* 9 ¿ 7
 When two fractions have the same denominators the one having a big numerator is a
greater one.
3 4
eg: * 5 ¿ 5

6 5
* 8 ¿ 8

9 6
* 10 ¿ 10

 For other fractions, before comparing them, first change them into decimal numbers.
3 2
eg: * 4 ¿ 5
0.75 0.4

1 1
* 4 ¿ 5
0.25 0.2

10. Ordering fractions

1) Arrange the following fractions from the largest to the smallest.
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 5 1 7 11
; ; ;
12 12 12 12
11 7 5 1
Solution: 12 ; 12 ; 12 ; 12

2) Re-arrange the following fractions is ascending order.

5 5 5 5 5
; ; ; ;
8 11 9 13 6
5 5 5 5 5
Solution: 13 ; 11 ; 9 ; 8 ; 6

3) Arrange the given fractions in ascending order.

2 3 1 5
; ; ;
3 4 2 6

2 3 1 5
Solution: 3 = 0.66.. 4
= 0.75 2
= 0.5 6
= 0.833
1 2 3 5
; ; ;
2 3 4 6
4) Arrange the following in descending order.
1 5
0.45 ; 3 ; 0.64 ; 6

11. Reciprocal of fractions

A reciprocal is a number that is related to another number by the fact that when the two
are multiplied their product is 1.
1 4
eg: - The reciprocal of 4 is 1 or 4
2 3
- The reciprocal of 3 is 2
5 8
- The reciprocal of 8 is 5
12. The complement of a fraction
A complement of a fraction is a fraction that can be added to that one to make a whole.
1 3
eg: - The complement of 4 is 4
2 1
- The complement of 3 is 3
13. Equivalent fractions
Equivalent fractions are fractions which are equal.
Examples
2
1) Write any two equivalent fractions to 3
2× 2 4
 3× 2 = 6

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 2× 3 6
 3× 3 = 9

1
2) Give any three fractions equivalent to 2
1× 2 2 1× 5 5
 =
2× 2 4
 2× 5 = 10

1× 3 3 1× 7 7
 2× 3 = 6  2× 7 = 14

3
3) Find the equivalent fraction to 5 whose numerator is 9.
3× 3 9
=
5× 3 15
9
The fraction is 15
4
4) Find the fraction equivalent to 7 whose denominator is 28.
4 × 4 16
=
7 × 4 28
16
The fraction is 28
2
5) Calculate the equivalent fraction to 5 whose sum of terms is 21.
Sum = 21
Sum of ratios = 2 + 5 = 7
1 ratio = 21 ÷ 7 = 3
Numerator = 2 x 3 = 6
Denominator = 5 x 3 = 15
6
The fraction is 15
4
6) Find the equivalent fraction to 7 whose difference of terms is 6.
Difference = 6
Difference of ratios = 7 – 4 = 3
1 ratio = 6 ÷ 3 = 2
Numerator = 4 x 2 = 8
Denominator = 7 x 2 = 14
8
The fraction is 14
14. Reducing fractions
Simplify completely:

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 60 2× 2× 3× 5 6
a) 70 = 2× 5× 7 = 7

45 3× 3 ×5 3
b) 75 = 3× 5 ×5 = 5

27 3 ×3 ×3 1
c) 81 = 3× 3 ×3 ×3 = 3

Exercises
Simplify completely:
36
a) 48 96 75
d) 144 g) 125
72
b) 108 180 90
e) 420 h) 135
90 100
c) 120 f) 125
60
i) 80

14. Operations in fractions

a) Addition and subtraction
When adding or subtracting two fractions with the same denominators, add or subtract
the numerators and keep one denominator.
Examples
Work out:
2 1 3
a) 5 +¿ 5 = 5

6 2 4
b) 7 −¿ 7 = 7

4 7 11 2
c) 9 +¿ 9 = 9 = 1 9

4 1 14 6 8 3
d) 2 5 −¿ 1 5 = 5 −¿ 5 = 5 = 1 5

3 1 19 9 28 4 1
e) 2 8 +1 8 = 8 +¿ 8 = 8 = 3 8 = 3 2

When adding or subtracting fractions with different denominators, first put them on the
common denominator.
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 Examples
Work out:
1 1 2+ 1 3
a) 2 + 4 = 4 = 4
5 3 20−9 11
b) 6 − 8 = 24 = 24

1 1 3+2 5
c) 4 + 6 = 12 = 12

1 1
d) 3 5 −1 2 =
16 3 32−15 17 7
− = = =1
5 2 10 10 10

1 1
e) 2 2 + 3 =
5 1 15+2 17 5
+ = = =2
2 3 6 6 6

1
f) 5 +2 =
1+ 10 11 1
= =2
5 5 5

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 Word problem involving addition and subtraction
1 1
1) Ana had 2 glass full of water and used 3 of it to take medicine. What fraction of water
remained?
1 3 1
2) In Amahoro stadium 6 of the seats is filled by women, 5 by men and 15 by children.
What fraction of the stadium is occupied?
1 1 2
3) 3 of the meeting room is filled by children, 5 by men and 5 by women. What fraction
of the seats in the meeting room are occupied?
13
4) A boy had a jerry can full of water, he used 20 . What fraction remained?
2 11
5) A tank was 3 full of water and after it rained, the tank was 12 full. What fraction was
added?
b) Multiplication
When multiplying a fraction by a fraction, multiply both numerators separately and then
denominators separately.
Examples
Multiply:
1 3
a) 3 × 4 = 4

1 2 2
b) 5 × 3 = 15

1 1 3 13 39 3 1
c) 1 2 × 2 6 = 2 × 6 = 12 =3 12 =3 4

Finding a fraction of a whole number

3
1) Find 4 of 200 kg.
3 3
4
of 200 kg = 4
×200 kg = 150 kg

2
2) What is 5 of 400m + 100m?
2
3) Calculate 3 of 3000g and express the answer in kg.
4) Find 0.25 of 3,000 Frw.
2
5) Ben had 10,000 Frw. He gave 5 of t to Gavin. How much money did Gavin get?
1 5 1
6) James, David and Tite shared 40,000 Frw. James got 4 , David got 8 and Tite got 8 .
How much money did each get?
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 Simplification
Simplify completely:
36 × 27 ×80 2× 2× 3× 3 ×3 ×3 ×3 × 2× 2× 2× 2× 5
a) 54 ×12 ×160 = 2× 3× 3 ×3 ×2 ×2 ×3 × 2× 2× 2× 2× 2× 5 =

18 ×32 ×21
b) 12× 64 × 42 =

30× 135 ×81

c) 27 × 40 × 45 =

15 ×9 × 42
d) 27 ×56 × 45 =

d) Division
When dividing a fraction by another fraction, multiply the first fraction by the reciprocal
of the second fraction.
Examples
Work out:
3 4 3 5 15
a) 4 ÷ 5 = 4 × 4 = 16

1 3
b) 2 3 ÷1 4 =¿

2
c) 5 ÷ 2=¿
1
3
3
d) 3 =¿
1
4
1
e) 54÷ 1 5 =¿

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UNIT 5: DECIMAL FRACTIONS/NUMBERS
1) The notion of a decimal
A decimal number is any number which contains a decimal point.
A decimal number also known as a decimal fraction is another way of expressing a
fraction.
Eg:
2
Two fifths = 5 = 0.4
2) Parts of a decimal number
A decimal numeral consists of three parts: - whole number
- decimal part
- decimal point
Eg: 0.5
Decimal part
Decimal point
Whole number
3) Place values
1) Write the place value of each digit in the number 23.0145
23.0145
Ten thousandths
Thousandths
Hundredths
Tenths
Ones
Tens
2) What is the rank of 6 in the number 0.2469?
3) What is the position of 4 in the number 726.9401?
4) Values
1) Find the value of each digit in the number 23.0145.
23.0145
5 x 0.0001 = 0.0005
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 4 x 001 = 0.004
1 x 0.01 = 0.01
0 x 0.1 = 0
3x1=3
2 x 10 = 20

2) What is the value of 9 in the number 0.00249?

3) What are the place value and the value of 3 in 45.936?
4) What is the rank of 4 and the lace value of 7 in the number 2483.9075?
5) Reading and writing decimal numbers
Write the following numbers in words:
a) 0.4 = Four tenths
b) 2.15 = Two hundredths
c) 3.142 = Three and one hundred forty to thousandths
d) 0.0126 = one hundred twenty six ten thousandths
e) 32.04 = Thirty two and four hundredths
f) 5.8 = Five and eight tenths
g) 0.26 = Twenty six hundredths
h) 573.075 = Five hundred seventy three and seventy five thousandths

6) Comparing decimal numbers

When comparing decimal numbers, first compare their whole parts.
Examples
Compare the following numbers:
a) 21.4 ¿ 7.1498
b) 6.1465 ¿ 8.1
c) 2.01 ¿ 3.1
d) 15.2 ¿ 9.64854
When whole parts are the same, compare decimal parts digits by digits
a) 2.4 ¿ 2.3987
b) 0.746 ¿ 0.8
c) 2.01 ¿ 2.01
d) 15.7 ¿ 15.64854
6) Ordering decimal numbers
1) Arrange the following numbers in descending order.
2.1 , 2.014 , 2.0 , 2.54
2.54 , 2.1 , 2.014 , 2.0

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2) Re-arrange the numbers below in ascending order.
3.7 , 3.08 , 3.2648 , 3.4
3.08 , 3.2648 , 3.4 , 3.7

7) Expressing decimals as fractions

Express the following decimals as fractions in their lowest terms:
45 9
a) 0.45 = 100 = 20

15 3 1
b) 1.5 = 10 = 2 =1 2

4 1
c) 0.04 = 100 = 25

45 9
d) 0.45 = 100 = 20

225 9 1
e) 2.25 = 100 = 4 =2 2

EXERCISES
Express the following decimals into fractions and simplify:
a) 0.5 f) 0.025
b) 0.75 g) 1.8
c) 0.625 h) 1.45
d) 20.4 i) 0.125
e) 3.75 j) 1.25
8) Rounding off decimal numbers
1) Round off 0.45 to the nearest tenths.
0.45
+1__
0.50 = 0.5
2) Correct 2.39246 to the nearest whole number.
2.39246
+0_____
2.00000 = 2
3) Correct 0.2485 to two decimal places.
0.2485
__+1__

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 0.2500 = 0.25

Exercises
1) Correct 34.728 to one decimal place.
2) Round off 0.493 to the nearest hundredths.
3) Correct 5.962 to the nearest tenths.
4) Round off 62.4896 to the nearest whole number.
5) Correct 0.2374 to two decimal places.

9) Operations in decimals
a) Addition
When adding decimals, arrange them vertically ensuring that the points are aligned.
Examples
Add: 1) 8.7501 + 21.2 =
8.7501
+ 21.2
29.9501

2) 23.96 + 6.4
23.96
+ 6.4
30.36
3) 6 + 2.45
6.00
+ 2.45
8.45

Examples
Add:
a) 6.3 + 4.5 f) 3.703 + 5.6
b) 10.6 + 5.31 g) 10.5 + 0.9
c) 3.5 + 17.285 h) 15.1 + 7.5
d) 3.83 + 28 i) 8 + 14.6
e) 7.1 + 8.21 j) 56.86 + 2.2

b) Subtraction
Arrange the numbers ensuring that the decimal points are aligned.
Examples
Subtract: 1) 46.4 – 8.2465 =
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 46.4000
−8.2465
38.1535
2) 17 + 3.72
17.00
−3.72
13.28
Exercises
Subtract:
a) 10.5 – 0.9 c) 45 – 9.4 e) 5.6 – 3.702 g) 0.7 – 0.005
b) 15.4 – 7.42 d) 41.5 – 7.52 f) 65.2853 – 8.95 h) 7 – 3.62

3) Multiplication
Multiply in ordinary, but the product must have the number of decimal places equal to
those in the multiplicand and multiplier.
Examples
Multiply:
a) 2.35 × 6.4
2.35
× 6.4
1410
+ 940
15.04
b) 26.856 × 8
26.856
×8
214.848 ¿
¿
Examples
Work out the following:
a) 6.5 × 1.2 c) 27 × 4.8
b) 7.5 × 0.16 d) 3.75 × 0.4
1) Quick multiplication by 10, 100, 1000, 10,000 etc
Multiply:
a) 9.264 × 100 = 926.4
b) 2.4 × 10 = 24
c) 7.5 × 100 = 750

4) Division
When dividing a decimal by a decimal, make the divisor a whole number.
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 Exercises
Work out:
8 4
a) 0.08 ÷ 4 = 0.02 or 0.08 ÷ 4 = 100 ÷ 1
8 1
0.02 = 100 × 4
2
4 0.08 = 100
0 = 0.02
00
0
08
8
0

2048 2
b) 20.48 ÷ 0.2 = 204.8 ÷ 2 = or 20.48 ÷ 0.2 = 100 ÷ 10
2048 10
102.4 = 100 × 2
10240
2 204.8 = 100
2 = 102.4
00
0
04
4
08
8
0
Exercises
Work out the following:
a) 1.2 ÷ 0.6 = l) 1.2 ÷ 0.6 =
b) 1.2 ÷ 0.6 =
c) 8.1 ÷ 0.027 =
d) 3.9 ÷ 0.03 = Quick division by 10, 100, 1000 etc
e) 48.8 ÷ 4 = Calculate:
f) 3636 ÷ 0.6 = a) 25.6 ÷ 10 = 2.56
g) 0.048 ÷ 0.12 = b) 784 ÷ 100 = 7.84
h) 0.204 ÷ 0.6 = c) 4000 ÷ 1000 = 4
i) 59.5 ÷ 0.07 = d) 45 ÷ 100 = 0.45
j) 1.2 ÷ 0.6 = e) 82.6 ÷ 10000 = 0.00826
k) 0.4 ÷ 0.002 =
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PRIMARY SIX MATHEMATICS Page 57 of 65 Tr. BATUZE JMV Tel: 0788768871
 TOPIC AREA: METRIC MEASUREMENTS
1. LENGTH MEASUREMENTS
Length is a distance between two points.
Measurements of length are the units used to measure the distance between two places.
The units of length are:
 Kilometer (Km)
 Hectometer (Hm)
 Decametre (Dam)
 Metre (M)
 Decimeter (Dm)
 Centimeter (Cm)
 Millimeter (Mm)
Conversion table of measurements of length.
Km Hm Dam M Dm Cm Mm
1 0
1 0
1 0
1 0
1 0
1 0

The measurements of area increase in the multiples of 10.

Note: The standard unit of length is metre (m)
Exercises
Complete:
1) 3 km = …………………….m
2) 3.7 dam = ………………km
3) 6 cm = …………………m
4) 2.5 m + 4 dam =…………………cm
5) 5.9 km – 12.6 dam = ……………………………m
6) 6 dam – 5 m = ……………………………hm
7) 36.87 dam + 5.83 m = ………………………hm
8) 3 km – 4 hm = …………………………m
2. MASS MEASUREMENTS
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Mass is the weight of the things contained by an object.
Measurements of mass are the units used to measure weight of things.
The units of mass are:
 Tonne (T)
 Quintal (Q)
 Kilogram (Kg)
 Hectogram (Hg)
 Decagram (Dag)
 Gram (g)
 Decigram (Dg)
 Centigram (Cg)
 Milligram (Mg)
Conversion table of measurements of mass.
t q - Kg Hg Dag g Dg Cg Mg
1 0
1 0
1 0
1 0 1
1 0 1 0
1 0 1 0

The measurements of area increase in the multiples of 10.

Note: The standard unit of length is kilogram (Kg)
Exercises
Complete:
1) 2 t = …………………….hag
2) 4.2 g = ………………mg
3) 6 hg = …………………q
4) 2.5 dag + 4 g =…………………cg
5) 3.9 dg – 5.3 mg = ……………………………g
6) 6 g – 5 dg = ……………………………hg
7) 36.87 q + 5.83 t = ………………………hg
8) 3 kg – 4 g = …………………………dag
3. CAPACITY MEASUREMENTS

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Measurements of capacity are the units used to measure the liquids.
The units of capacity are:
 Kilolitre (Kl)
 Hectolitre (Hl)
 Decalitre (Dal)
 Litre (l)
 Decilitre (Dl)
 Centilitre (Cl)
 Millilitre (Ml)

Conversion table of measurements of capacity.

Kl Hl Dal L Dl Cl Ml
1 0
1 0
1 0
1 0
1 0
1 0

The measurements of area increase in the multiples of 10.

Note: The standard unit of capacity is litre (l)
Exercises
Complete:
1) 6 dal = …………………….cl
2) 7 l = ………………kl
3) 5.3 dl = …………………ml
4) 4.6 cl = ……………….dal
5) 1.2 hl – 2.5 dal = ……………ml
6) 3.9 dal + 2.8 l = ……………….cl
7) 3 kl – 6 dal = ……………….hl
8) 56.8 l + 2.4 dl = ………………….ml

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 4. MEASUREMENTS OF AREA OR SURFACE MEASUREMENTS.
Measurements of area are the units used to measure the area.
The units of area are:
 Square Kilometer (Km2)
 Square Hectometer (Hm2)
 Square Decametre (Dam2)
 Square Metre (m2)
 Square Decimeter (Dm2)
 Square Centimeter (Cm2)
 Square Millimeter (Mm2)

Conversion table of surface measurements.

Km2 Hm2 Dam2 m2 Dm2 cm2 cm2

1 0 0
1 0 0
1 0 0
1 0 0
1 0 0
1 0 0
The
measurements of area increase in the multiples of 100.
Note: The standard unit of length is square metre (m2)
Exercises
Complete:
a) 3 km2 = ……………………. m2
b) 3.7 dam2 = ………………k m2
c) 6 cm2 = ………………… m2
d) 2.5 m2 + 4 dam2 =…………………cm2
e) 5.9 km2 – 12.6 dam2 = …………………………… m2
f) 6 dam2 – 5 m2 = ……………………………hm2
g) 36.87 dam2 + 5.83 m2 = ………………………hm2
h) 3 km2 – 4 hm2 = ………………………… m2

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 5. LAND MEASUREMENTS
Land measurements are the measurements used to calculate the area of a land.
The land units are:
 Hectare (ha)
 Are (a)
 Centare (ca)

Conversion table of units of land

Ha a Ca
1 0 0

1 0 0

The measurements of land increase in the multiples of 100.

Exercises
Complete:
1) 4 ha = ……………………. a
2) 4.8 a =………….………..ca
3) 2.4 a = …………………….ha
4) 3.6 ha + 3.7 a = ……………….ca
5) 12.5 a – 3.8 ca = ………………..ha

RELATIONSHIP BETWEEN SURFACE AND LAND MEASUREMENTS

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Consider: 1 square hectometer = 1 hectare
1 hm2 = 1 ha
Conversion table

Km2 Hm2 Dam2 m2 Dm2 cm2 cm2

ha a ca
1 0 0
1 0 0
1 0 0
1 0 0
1 0 0
1 0 0

Exercises
Work out:
1) 36 ha = ………………….dam2
2) 2.9 m2 = ………………………a
3) 3.6 km2 – 3 ha = ……………………….m2
4) 6 dm2 + 4 m2 = ………………………a
5) 12.7 hm2 – 45 m2 = ……………………….a
6) 125 ha + 36 dam2 = ……………………….cm

GEOMETRIC FIGURES
1. SQUARE
A square is a figure that has four equal sides and four right angles.

Properties of a square
 It has four equal sides
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  It has four right angles
 It has two equal diagonals
 It has four lines of symmetry

Perimeter

Perimeter means the distance around a figure

Perimeter of a square = S + S + S + S
P=S×4
EXAMPLES
1) Find the perimeter of a square whose side is 15 cm
S = 15 cm
P=SX4
=15 cm x 4 = 60 cm
2) The side of a square garden is 40 m. calculate its perimeter.
3) Manzi’s square farm has a side of 80 m. calculate the perimeter of the farm
4) The length of side of a square is 65 cm. find its perimeter.

Finding the side

1) The perimeter of a square is 40 cm. Find the side.
P = 40 cm
S = P ÷ 4 = 40 cm ÷ 4 = 10 cm
2) Calculate the length of side of a square whose perimeter is 100 m.
3) The distance around a square garden is 180 m. Find the side.
4) Find the side of a square whose perimeter is 84 dm.

Area
1 2 3 4 5 The area is the surface covered by the boxes
2 Area = number of boxes of one side x that of another side
3 Area = 5 x 5
4 Therefore, A = S X S
5

EXAMPLES
1) The side of a square is 12 cm. Find the area.
PRIMARY SIX MATHEMATICS Page 64 of 65 Tr. BATUZE JMV Tel: 0788768871
 S = 12 cm
A=SXS
= 12 cm x 12 cm = 144 cm2
2) The perimeter of a square is 36 cm. Find the area.
P = 36 cm
A=SXS
S = P ÷ 4 = 36 cm ÷ 4 = 9 cm
A = 9cm x 9 cm = 81 cm2
3) Find the area of a square whose side is 45 cm.
4) The perimeter of a square garden is 160 m. Find the area.

PRIMARY SIX MATHEMATICS Page 65 of 65 Tr. BATUZE JMV Tel: 0788768871