1st SEMESTER
BUSINESS MATH-GRADE 11
Learning Module No. 3
Name: _________________________________ Date: ___________________
Grade level & Section: __________________ Score: __________________
Name of Subject Teacher: NISSAN A. TOSTON Contact Number: 09774675979
Topic: KEY CONCEPTS OF RTIO AND PROPORTION
Duration: 2 Weeks
Learning Competencies: 1. Identify the different kinds of proportions and write examples of real-life situations for
each
2. Solve the problems involving direct, inverse and partitive proportion
Learning Objectives: Students were expected to:
1. Compare and differentiate real-life problem
2. Write proportions illustrating real-life problems
3. Identify, describe and give examples of the different types ofproportion
4. Solve problems involving direct, inverse, and partitive proportions
References: Business Math for Engaged Learning pp.45-68
INTRODUCTION
A ratio is a comparison of the sizes of two or more quantities. The common forms of expressing ratios are the
following: in fraction form, using a colon to separate the quantities, or the use of the word “to”.
A statement expressing the equality of two ratios is called a proportion. Hence, if the ratios a:b and c:d are equal,
a c
then the equation = is a proportion. The numbers b and c are called the means, while the numbers a and d are called
b d
the extremes. If we multiply the two ratio in the above proportion by the product bd of their denominators, then we get the
equation a × d = b × c. This shows the Extremes-Means Property of proportions, which states the following:
Extremes-Means Property of Proportions:
a c
If = is a proportion and b=0, d=0, then the product of the means, b × c, id equal to the product of the
b d
extremes, a × d.
Since the products a × d and b × c are obtained by diagonal multiplying the terms of the two ratios when these
ratios are expresses fraction form, the two products are also referred to as the cross products of a proportion, and the
above property says that when two ratios in fraction form are in a proportion, then the two cross products are equal.
PRE-TEST
Find the ratio of the following problems and determine the ratio if it forms a proportion.
1. In a class of 40 students, 24 are girls and 16 are boys. What is the ratio of boys to girls?
2. In a class of 40 students, 24 are boys and 30 are girls. What is the ratio of boys to girls?
35 20
3. a. Determine if the ratios and form a proportion.
91 52
3 5
4. Solve for x in the proportion =
x x+ 4
Excellent! You did a good job on that task. This time, read “The Content of
the Topic” to know more about the characteristics, processes and ethics of
research.
CONTENT
Thus, for example, if a class consists f 24 boys and 30 girls, then the ratio of boys to girls can be written in any of
the following forms:
24 4
= , 4:5. 4 is to 5
30 5
The numbers that belong to a given ratio are called the terms of the ratio. When a ratio involves more than two quantities,
the colon form is used. For example, the ratio of the three numbers 5, 3, and 2 can be written as 5:3:2 and read as “5 is to
3 is to 2”. When writing ratios, the units of measurement are excluded. However, the numbers in a ratio must be in the
same units. For example, the ratio of 45 minutes to two hours can be expressed by first converting two hours to 2 x 60 =
120 minutes and ten writing the ratio as:
45 3
= or 3:8.
120 8
Example 1.1
a. In a class of 40 students, 24 are girls and 16 are boys. The ratio of boys to girls is 16:24 or 2:3, the ratio of girls to the
total number of students in the class is 24:40 or 3:5.
b. b. In a conference, the number of European, American and Asian participants are 60, 72, and 48, respectively. Hence,
the ratio of European, American, and Asian participants is 60:72:48, or 5:6:4., in the simplest form.
When the quantities are being compared cannot be expressed in the same unit, then the number used to compare
them is called rate instead of a ratio. The following are examples of rates:
Example 1.2
a. A proof reader finds that a certain manuscript contains an average of 3 errors per page
b. A car travel at an average speed of 60 kilometers per hour.
c. A car consumes gasoline at the rate of 10 kilometers per liter
d. A recipe calls for 2 eggs for every 1.5 cups of flour
e. A painter consumes paint at the rate of one gallon of paint for every 50 square feet of surface
Two ratios are said to be equivalent if their fraction representations are equivalent fractions. Hence, two ratios a:b
and c:d are equivalent if and only if a x d = b x c. A ratio is said to in simplest form if gcf(a,b) = 1.
Example 1.3
a. The ratios 4:7 and 12:21 are equivalent ratios, since 4x21 = 7x12 = 84. Moreover, when written as fractions, they
4 12
represent equivalent fractions, since =
7 21
1 1
b. To express the ratio : into an equivalent ratio with integer terms and in simplest form, we may multiply both terms
4 2
by the least common denominator of the two fractions. This gives
1 1
4×
4 4 1
= =
1 1 2
4×
2 2
Which shows that the simplest form of the ratio is 1;2.
c. In an outreach program, students in Grade 11 class went to an orphanage to spend a day with the children in the center.
Two students were tasked to take care of the children in the group activities and games. If 40 Grade 11 students went to
the said program and the above ratio was followed, then if n is the number of children who participated in the activities, we
have the equivalent ratios 2:5 and 40:n, so that 2 x n = 5 x 40 = 200, and so n = 100 is the number of children in the
orphanage who participated in the activities.
d. A man divided hi estate in the ratio of 5:2:2:1 for his wife, his two children and to charity. If the money given to charity
amounted to1.5 million pesos, find the total amount of his estate, and the amount received by each of his two children.
Solution: Since the value of the estate was divided into ten equal parts, the money that was given to charity represents
one of these equal parts. If we let x be the amount of the estate, then:
10 x 1.5 = x = 15
Which means that the total amount f the estate is 15 million pesos, while each of his two children received 2 x 1.5 = 3
million pesos.
e. At a dry goods market, one store sells eggs at 5 pieces for P28, while the store next to it sells eggs at the same size at
P63 per dozen. Which store sells eggs at a cheaper price?
Solution: In the first store, the ratio of the price to eggs is 28:5, which means that each egg costs P5.60. in the second
store, the corresponding ratio is 63:12 = 21:4, which shows that the eggs would sell at P5.25 per piece. Hence, the second
store is selling the egg at a cheaper price.
f. The usual mix of concrete is one part cement, to parts sand, and three parts gravel. If 60 kilograms of concrete mixture
are needed, how much cement, sand and gravel must be used?
Solution: Let x be the size of each part. Thus, the mixture consists of six equal parts, or 6x. This gives us:
6x = 60 x = 10
This shows that in order to obtain the required amount of mixture,
X = 10 kilograms of cement
2x = 20 kilograms of sand
3x = 30 kilograms of gravel must be used.
If we multiply the two ratio in the above proportion by the product bd of their denominators, then we get the
equation a × d = b × c. This shows the Extremes-Means Property of proportions, which states the following:
Extremes-Means Property of Proportions:
a c
If = is a proportion and b=0, d=0, then the product of the means, b × c, id equal to the product of the
b d
extremes, a × d.
Since the products a × d and b × c are obtained by diagonal multiplying the terms of the two ratios when these
ratios are expresses fraction form, the two products are also referred to as the cross products of a proportion, and the
above property says that when two ratios in fraction form are in a proportion, then the two cross products are equal.
Example 1.1
35 20
a. Determine if the ratios and form a proportion.
91 52
Solution:
If the two ratios form a proportion, then the Extremes-Means Property of Proportions, the cross products should be equal.
We have:
35 × 52 = 1.820, 91 × 20 = 1,820
35 20
which shows that the cross products are equal. Hence, the two ratios form a proportion, and we can write = .
91 52
Another way of determining whether the two ratios are in a proportion is to express them both in simplest form. We have:
35 7× 5 5 20 4 × 5 5
= = , = =
91 7 ×13 13 52 4 × 13 13
Since the two ratios have the same simplest form, they must be equivalent, and hence form a proportion.
3 5
b. Solve for x in the proportion =
x x+ 4
Solution:
Using the extremes-means property of proportions, we equate the cross products to get
5x = 3(x+4) = 3x + 12 2x = 12, x = 6
c. A man wants to invest some of his savings in a conservative portfolio where the ratio of the amount investment in bonds
to the amount invested in stocks is 3 is to 1. If the amount invested in bonds is P250,000 more than the amount invested in
stocks, determine the amount of each investment.
Solution:
If the amount invested in stocks is denoted by x, then the amount invested in bonds is 3x, and we have:
3x = x = 2x = 250,000 x = 125,000
Hence, the man invested x = P125,000 in stocks and 3x = P375,000 in bonds.
3 types of Proportions:
1. Direct Proportions
We consider two quantities, say x and y, that are related in such a way that if the value of x changes, then the
y
value of y also changes, such that the ratio is a constant, say k. Thus, the values of x and y are related by n equation of
x
the form y = kx, k ≠ 0. In this case, we can say that y and x are directly proportional, or that y is directly proportional to x, or
that y and x re in a direct proportion. The number k is called the constant of proportionality. A direct proportion is also
called a direct variation, and the constant k is al known as the constant of variation.
Example 1.1
a. The circumference of a circle can be computed from the formula C = 2πr, where C is the circumference and r denotes
the radius. We see that the circumference is directly proportional to the radius and the constant of proportionality is k = 2π.
27 9
b. If the eggs are sold at P27.00 for 6 pieces, then the ratio of the price to the number of eggs is = = 4.5 and y =
6 2
4.5x, where x is the number of egg and y is the cost in peso of x eggs. Here, 4.5 represents P4,50, which is the price of
one egg. Using the above equation, the cost of 2 dozen eggs is given by
y = 4.5x = 4.5 x 24 = 108
Since two dozens is equivalent to 2 x 12 = 24 pieces. Hence, 24 eggs would cost P108.
c. Revenue from sales of a single item is computed by multiplying the number x of units sold by the unit selling price p. If
y
the total revenue from sales is denoted by y, then we can write y = px = , which shows that revenue is directly
x
proportional to the number of units sold, with the unit selling price p as the constant of proportionality.
Inverse Proportions
When two quantities x and y are related in such a way that the product of their values is a constant, that is, there exists a
nonzero constant k product so that xy = k, then we say that x and y are inversely proportional to k such each other, or that
x and y are in an inverse proportion. If k is positive, then the value of y increases as x decreases, and vice versa. Inverse
proportions are also called inverse variations.
Example 1.1
In Chemistry, Boyle's Law states that an ideal gas of given mass exerts pressure P which is inversely proportional
to the volume of space V occupied by the gas. In equation form, Boyles law is given by PV =k.
Example 1.2
A common example of two quantities which are inversely proportional to each other is the number of workers
working together on a task and the length of time it takes to finish the task, assuming that the workers all have the same
rate. In the same manner, another application follows.
Example 1.3
Suppose it takes four pipes 70 minutes to fill a water tank, how long will it take to fill the tank if seven pipes are used?
-
Solution:
An increase in the number of pipes used will result to a decrease in the amount of time needed to fill the tank. Thus, the
number of pipes used and the amount of time to do the task are in inverse proportion, and the product of their values is a
constant. If we let x represent the unknown time, then we have
280
4x70 7x>7x = 280, x = = 40
7
which shows that it will take 40 minutes to fill the tank with water using seven pipes.
Example 1.4
Example 2.
If a car salesman drives at a constant rate of 60 kilometers per hour, it takes him 2.5 hours to drive from one city to
another. How long will it take him to cover the same distance if he were to drive at a constant rate of 80 kilometers per
hour, assuming the same road conditions?
Solution:
Distance, rate, and time are related by the equation d= r x t. Since the distance to be covered is the same, the rate r and
the time I are inversely proportional. Hence, we have
150
2.5x60= tx80 80t = 150, t = = 1.875 hours.
80
Partitive Proportions
When a quantity is divided into two or more parts such that the parts are in a definite ratio with each other, then the
quantities representing the parts are said to be in a partitive proportion. To solve problems involving partitive proportions,
we usually equate the sum of the parts to the whole quantity. This is illustrated in the following examples.
Three investors investėd a total amount of P 765,000 in a business in the ratio 2:3:4. How much was the biggest
investment?
-Solution:
The entire amount may be divided into nine equal parts. If we let x denote the amount of each part, then the amounts
invested may be represented by 2x, 3x, and 4x, respectively. This gives us
765 ,000
2x+3x+ 4x = 9x =765,000 x= = 85,000
9
so that the biggest amount invested is 4x = 4x85,000 = P340,000
Example 1.1
In the preceding example, suppose that the profit from the size investment was divided among the three investors
according 00 of their investment. If the smallest investor received a profit P 55,000, how much was the total profit?
-Solution:
As with the investment, the profit may be divided into nine equal parts, and the smallest profit represents two of these
parts. If we let x represent one of the equal parts, then we have:
2x=55,000 x = P27,500
and the total profit is 9x = 9 x 27,500 = P247,500.
Example 1.2
A man plans to donate his collection of 3,042 books to three libraries in the ratio 1:3:5. How many books will each library
receive from his donation?
Solution:
The numbers of books that the three libraries will receive may be represented by x, 3x, and 5x, respectively, so that
3,042
x+3x +5x = 9x =3,042 x= = 338. This shows that the three libraries will receive 338 books, 3x338 = 1,014
9
books, and 5x338 1,690 books, respectively.
Example 1.3
At a movie premiere, the number of children, men, and women who attended were in the ratio 2:3:4. If 240 children
watched the premiere, how many persons were in the audience?
Solution:
The number of persons who attended the premiere can be divided into nine equal parts, and two of these parts are made
up of children. If x represents one part, then we have
2x 240 x =120.
Thus, the total number of persons in the audience who attended the premiere is 11x =11x120=1,320.
Joint and Combined Variations
Aside from direct, inverse, and partitive proportions or variations, there are two other types of variations which may appear
less frequently than the first three types: these are joint variations and combined variations. If one quantity is directly
proportional to the product of two or more other quantities, then the quantities involved are said to be jointly proportional, or
are in a joint variation. On the other hand, if one quantity varies directly with one or more quantities, and inversely with at
least one other quantity, then the quantities are said to form a combined variation.
-Example 1.1
The kinetic energy of an object in motion is given by the formula
1 2
KE = mv
2
where KE is the kinetic energy, m is the mass of the object, and v is its velocity. The above formula tells us that the kinetic
energy of a moving object is jointly proportional to its mass and the square of its velocity, with as the constant of
proportionality.
Example 1.2
Distance, rate, and time are related by the equation d = rt, where d is the distance, r is the rate or speed, and t is the time.
We see from the given formula that distance varies jointly with rate and time.
Example 1.3
Two objects in the universe exert gravitational force on each other. From Newton's laws, we know that the force due to
gravity exerted by the first object on the second is the same as the force exerted by the second object on the first object. If
we denote this force by F g3, then the formula for computing this force is:
G x m1 x m 2
Fg =
d
where G is the universal gravitational constant, m, is the mass of the first object, m, is the mass of the second object, and d
is the distance between the two objects. The above formula tells us that the force due to gravity varies jointly with the mass
of the objects and inversely with the square of the distance between the two objects. This shows that the variables in the
above formula are related by a combined variation.
Example 1.4
Suppose it is known that a quantity z varies directly with a second quantity x and inversely with the product of two
quantities y and v.
a. Find the equation that defines the variation.
b. If z =5 when x = 30, y ==12, and v=10, find the constant of proportionality. Then, determine the value of x when
V=10, y=2, and z =12.
Solution:
kx
a. Since z varies directly with x and inversely with the product of y and v, we have the equation z = where k is the
yv
constant of proportionality.
b. Substituting the given values into the equation in (a), we have
30 k
5=
( 12 )( 10)
k
5=
4
20 = k
The specific formula defining the variation is z = Hence, for the second part, we just substitute the given values to obtain
the value of x, thus
20 x
5=
( 2 ) (10)
12 = x Or
X = 12
Enrichment Activity
ACTIVITY 1. Express the following ratio in simplest form with integer terms.
2.5
a. =
3.5
0.05
b. =
0.8
2
3
c. =
3
4
1
1
2
d.
2
2
3
12.5 %
e. =
0.375
ACTIVITY 2: Solve the following problems using ratios.
a. If a photocopying machine can make 40 copies in one minute, how many copies can it make in 8.4 minutes, assuming
that it operates at the same rate?
b. If five oranges cost P60, how much will a dozen oranges cost?
ACTIVITY 3: Determine if the given ratios form a proportion.
a. 5:9 and 135 : 405
b. 5 : 12 and 7 : 15
c. 2.5 : 3.5 and 15% : 22%
d. 0.45 : 0.6 and 60% : 80%
ACTIVITY 4: Show that the cross products are equal.
18 36
a. = =
23 46
10.5 12.6
b. = =
250 300
1.6 0.96
c. = =
1.7 1.02
d. 8 : 0.77 = 32 : 3.08 =
Generalization
In this section, you studied different types of proportions or variations. Did you have any difficulty in differentiating
among these different types of variations? If you see a formula that describes a proportion, how can you determine the type
of proportion described by the formula?
I thought…
I learned…
Prepared by: Checked by:
NISSAN A. TOSTON MA. RESA M. GALIDO
Subject Teacher Subject Area Coordinator
Noted by:
MIRA ROCHENIE C. PIQUERO
Academic Coordinator
Approved by:
MA. NANETTE D. BAYONAS
OIC-School Principal
