Math-E-Matics Classes

FRACTIONS

CONVERSIONS

A. Changing a Mixed Number to an Improper Fraction

2
Mixed number – 4 (contains a whole number and a fraction)
3
14
Improper fraction - (numerator is larger than denominator)
3

Step 1 – Multiply the denominator and the whole number

Step 2 – Add this answer to the numerator; this becomes the new numerator
Step 3 – Carry the original denominator over

1
Example #1: 3 = 3 × 8 + 1 = 25
8

25
8

4
Example #2: 4 = 4 × 9 + 4 = 40
9

40
9

B. Changing an Improper Fraction to a Mixed Number

Step 1 – Divide the numerator by the denominator

Step 2 – The answer from step 1 becomes the whole number
Step 3 – The remainder becomes the new numerator
Step 4 – The original denominator carries over

9
47 2
Example #1: = 47 ÷ 5 or 5 47 = 5 47 = 9
5 5
45

2
4
9 1
Example #2: = 2 9 = 2 9 = 4
2 2
8
1

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C. Reducing Fractions

Step 1 – Find a number that will divide into both the numerator and the
denominator
Step 2 – Divide numerator and denominator by this number

10 2
Example #1: = (because both 10 and 15 are divisible by 5)
15 3

4 1
Example #2: = (because both 4 and 8 are divisible by 4)
8 2

D. Raising Fractions to Higher Terms When a New Denominator is Known

Step 1 – Divide the new denominator by the old denominator

Step 2 – Multiply the numerator by the answer from step 1 to find the new
numerator
*Note: If the original number is a mixed number, convert it to an improper
fraction before raising to higher terms (see Example #2)

2 2 8
Example #1: = becomes = because 12 ÷ 3 = 4
3 12 3 12
and 2 × 4 = 8

1 11 11 44
Example #2: 2 = becomes = becomes =
5 20 5 20 5 20

because 20 ÷ 5 = 4 and 11 × 4 = 44

E. Converting fractions to decimals

Step 1 - Divide the numerator (the top number) by the denominator (the
bottom number) of the fraction.

Example: 5
8
.625
8 5.000  Add as many zeros as needed.
48
20
16
40
40
0

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F. Converting decimals to fractions

Step 1 - Determine the place value of the last number in the decimal; this
becomes the denominator.
Step 2 – Make the decimal number your numerator.
Step 3 - Reduce your answer.

Example: .625 - the 5 is in the thousandths column, therefore,

625 5
.625 = = reduces to
1000 8

(Hint: Your denominator will have the same number of zeros as there are decimal
digits in the decimal number you started with - .625 has three decimal digits so the
denominator will have three zeros before reductions.)

G. Multiplying Simple Fractions

Step 1 – Multiply the numerators

Step 2 – Multiply the denominators
Step 3 – Reduce the answer to lowest terms

1 4 4 2
Example: × = which reduces to
7 6 42 21

H. Multiplying Mixed Numbers

Step 1 – Convert the mixed numbers to improper fractions first

Step 2 – Multiply the numerators
Step 3 – Multiply the denominators
Step 4 – Reduce the answer to lowest terms

1 1 7 3 21 1
Example: 2 × 1 = × = which then reduces to 3
3 2 3 2 6 2

*Note – When opposing numerators and denominators are divisible by a common number, you may
reduce the numerator and denominator before multiplying. In the above example, after converting the
mixed numbers to improper fractions, you will see that the 3 in the numerator and the opposing 3 in the
denominator could have been reduced by dividing both numbers by 3, resulting in the following
reduced fraction:

7 31 7 1
× = = 3
13 2 2 2

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I. Dividing Simple Fractions

Step 1 – Change division sign to multiplication

Step 2 – Change the fraction following the multiplication sign to its
reciprocal (flip the fraction around so the old denominator is the
new numerator and the old numerator is the new denominator)
Step 3 - Multiply the numerators
Step 4 – Multiply the denominators
Step 5 – Change the answer to lowest terms

1 2 1 3 3
Example: ÷ = becomes × which when solved is
8 3 8 2 16

J. Dividing Mixed Numbers

Step 1 – Convert the mixed number or numbers to improper fraction

Step 2 – Change the division sign to multiplication
Step 3 – Change the fraction following the multiplication sign to its
reciprocal (flip the fraction around so the old denominator is the
new numerator and the old numerator is the new denominator)
Step 4 - Multiply the numerators
Step 5 – Multiply the denominators
Step 6 – Change the answer to lowest terms

3 5 15 17 15 6
Example: 3 ÷ 2 = becomes ÷ becomes × =
4 6 4 6 4 17

15 63 45 11
which when solved is × = which simplifies to 1
24 17 34 34

K. Adding and Subtracting Fractions

Step 1 – Find a common denominator (a number that both denominators

will go into)
Step 2 – Raise each fraction to higher terms as needed
Step 3 – Add or subtract the numerators only as shown
Step 4 – Carry denominator over
Step 5 – Change the answer to lowest terms

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1 7
Example #1: + = Common denominator is 8 because both 2 and
2 8
8 will go into 8
1 4
=
2 8
7 7
+ =
8 8

11 3
which simplifies to 1
8 8

3 1
Example #2: 4 – = Common denominator is 20 because both 4
5 4
and 5 will go into 20

3 12
4 = 4
5 20
1 5
– =
4 20

7
4
20

1 1 1 8 9
Example #3: 2 = 2 = 12 + = 1
8 8 8 8 8
1 2 2 2
– 1 = 1 = 1 = 1
4 8 8 8

7
**
8
**Note – In this problem you must borrow from the whole number to adjust your fraction so
that you can subtract. However, you may do this problem another way. Simply change the
mixed number to improper form before finding the common denominator to prevent having to
borrow.

1 17 17
=2 =
8 8 8
1 5 10
– 1 = =
4 4 8

7
8

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Comparing Fractions

If Denominator is same then compare the numerator if numerator is greater then

the fraction is greater if numerator is small then fraction is least one.
For Example
12 6
and
13 13
From above denominator is 13 then compare 12 and 6:- (12> 6)
12 6
Then >
13 13
If Numerator is same then compare the denominator if denominator is greater
then the fraction is smaller if denominator is small then fraction is greatest one.

For Example
12 12
and
13 17

Numerator is same then we compare denominator 13< 17 then

12 12
>
13 17

If neither denominator nor numerator is same then we do it by cross multiplication

For Example
12 11
and
13 17

Cross multiply each other

12 x 17 = 204 13 x 11 = 143
204 > 143
Therefore
12 11
>
13 17

If the difference between the numerator and denominator of two or more proper fractions is
same then the fraction with higher numerator is greatest and lowest numerator is smallest.

If the difference between the numerator and denominator of two or more improper fractions is
same then the fraction with higher numerator is Lowest and lowest numerator is Greatest.

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FRACTIONS PRACTICE SHEET

A. Write as an improper fraction.

1 1 2 3
1. 1 2. 4 3. 1 4. 2
8 5 3 16

5 1 5 4
5. 2 6. 2 7. 1 8. 3
7 16 8 5

1 2 5 1
9. 7 10. 5 11. 3 12. 6
4 3 6 2

B. Write as a mixed number.

10 19 25 9
1. 2. 3. 4.
4 2 3 8

25 35 7 21
5. 6. 7. 8.
16 4 3 8

4 12 17 48
9. 10. 11. 12.
2 7 4 9

C. Write in lowest terms.

6 21 18 12
1. 2. 3. 4.
32 35 24 15

5 9 14 8
5. 6. 7. 8.
30 27 49 32

12 16 8 10
5. 1 10. 2 11. 5 12. 3
21 20 14 25

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D. Find the missing numerator by raising the fraction to higher terms.

3 ? 7 ? 5 ? 5 ?
1. = 2. = 3. = 4. =
4 12 16 64 8 48 9 72

2 ? 4 ? 1 ? 3 ?
5. 5 = 6. 1 = 7. 1 = 8. 2 =
3 12 5 10 4 12 5 10

E. Convert the following fractions into decimals.

2 1 4 5 7 9
1. 2. 3. 4. 5. 6.
3 8 5 6 16 16

F. Convert the following decimals to fractions.

1. .225 2. .375 3. .0175 4. .95 5. .5 6. .45

G. Multiply.

1 1 7 2 3 2 1 3
1. × = 2. × = 3. × = 4. × =
9 2 10 5 8 7 2 16

3 2 7 4 15 1 2 5
5. × = 6. × = 7. × = 8. × =
4 3 16 3 64 12 9 9

3 1 5 3 5 3
9. × 10 = 10. 1 × = 11. × = 12. 14 × =
4 2 6 16 12 8

1 1 1 1 1 1
13. × 1 = 14. 3 × = 15. 18 × 1 = 16. 16 × 2 =
2 3 16 5 2 8

3 3 2 3 4 2 1 2
17. 6 × 1 = 18. 2 × 4 = 19. 4 × 4 = 20. 3 × 2 =
8 5 3 8 9 4 8 5

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H. Divide as shown.

1 1 2 1 8 2 2 1
1. ÷ = 2. ÷ = 3. ÷ = 4. ÷ =
2 4 5 2 3 3 9 3

1 4 3 6 4
5. 4 ÷ = 6. 8 ÷ = 7. 9 ÷ = 8. ÷ =
8 5 4 5 5

4 1 2 5 2 7
9. ÷ = 10. ÷ = 11. ÷ 4= 12. 14 ÷ =
11 11 7 9 3 8

5 3 1 1 1
13. 15 ÷ = 14. 8 ÷ = 15. 1 ÷1 = 16. 3 ÷ 5 =
6 4 4 2 2

1 1 1 2 3 1 1 5
17. 6 ÷ 2 18. 5 ÷ 2 = 19. 2 ÷ 1 = 20. 3 ÷1 =
4 2 3 3 4 8 5 7

I. Add or subtract as shown.

3 7 2 3 3 1 3 5
1. + = 2. + = 3. + = 4. + =
8 8 3 4 32 8 5 6

5 1 3 1 1 1 1 1
5. + = 6. + 1 = 7. + = 8. 2 + 1 =
8 10 8 4 4 5 8 4

5 13 2 4 9 3 7 1
9. 1 + = 10. 2 + = 11. – = 12. – =
8 16 3 9 10 16 8 2

11 1 5 1 7 3 1 3
13. – = 14. – = 15. – = 16. 1 – =
16 4 6 5 8 10 2 32

5 3 2 7 1 5 5 1
17. 5 – 2 = 18. 3 – 1 = 19. 2 – = 20. 4 – 1 =
6 9 3 8 4 6 6 2

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Solve the word problems below using fractions.

1. The Cooper family decided to hike to Hillside Lake, approximately 8⅝ miles away. After
an hour the lake was still 5⅓ miles away. How far did the group hike so far?

2. While riding her bike, Susan burns 450 calories every ½ hour. Based on this rate, how many
calories will Susan burn if she rides the bike for 1¾?

3. Last Friday Tony worked for 7½ hours. Express this time as a fraction of the day.

4. When an oil tank is 7

12 full, it contains 5¼ gallons. How many gallons does it hold when
full?

5. How many pieces of 10 516 inch bar can be cut from a stock 20 foot bar?

6. Byron purchased a box of candy at the store. On his way home he ate ¼ of the candy in the
box. At dinner with friends later that night he served ½ of what was left. If there are 6
chocolates now left in the box, how many did the box contain to start with?

7. Seth earns $560 per week. He has 15 of his income withheld for federal taxes, 115 of his
income withheld for state taxes, and 1 25 of his income withheld for medical coverage. How
much income is left each week after those deductions?

8. A bolt extends through ¾” thick plywood, a washer that is ⅛” thick, and a nut that is 316 ”
thick. The bolt should be ⅝” longer than the sum of the thickness of the plywood, washer
and nut. What is the minimum length of the bolt?

9. A recipe for French toast that serves 6 calls for ¾ cup granulated sugar, 1 cup of evaporated
milk, ⅓ teaspoon of vanilla, and 12 thick slices of French bread. How much of each
ingredient is needed to serve only three?

10. Boll’s Electrical has a washing machine on sale for ⅓ off the regular price of $429. What is
the sale price of the washing machine?

11. For a family party, Tanisha made 2 5 of the desserts. If a total of 40 desserts were brought to
the party, how many did Tanisha supply?

12. The price of computers has fallen by 2

5 . If the price of a computer was originally $10,275,
by how much has the price fallen?

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