Unit 3
RATIO AND PROPORTION
LESSON OBJECTIVES
 Define ratio and proportion
 Perform the techniques in solving ratio and proportion
 Deepen the knowledge in ratio and proportion by solving different applications
About Ratios
Things around us can be paired or compared if they are of the same kind and can be
expressed mathematically through what we called ratio. Ratio is defined as comparison of two
quantities, usually expressed as a quotient. The ratio of “x to y” can be denoted by “x : y or x/y”
since ratio is a quotient, then y ≠ 0.
Ratios can be reduced to its simplified form. For instance, you want to compare the
interest of your money in the bank as to the number of years “2000 : 5 or 2000/5” can be
simplified as “400 : 1 or 400/1”.
Example 1:
Simplify the given ratios.
1. 12 cm : 2 m
2. 100 m : 2 km
Solution:
Since the given ratios contain different units, we can convert them to like units
and simplify or reduce the ratio as possible.
1.
12 𝑐𝑚
2𝑚
=
2.
100 𝑚
2 𝑘𝑚
=
12 𝑐𝑚
2𝑚𝑥
100 𝑐𝑚
1𝑚
=
100 𝑚
2 𝑘𝑚 𝑥
1000 𝑚
1 𝑘𝑚
12 𝑐𝑚
200 𝑐𝑚
=
=
100 𝑚
2000 𝑚
12 ÷4
200 ÷4
=
=
3
50
100 ÷100
2000 ÷100
=
1
20
Applying the Ratios
1. The perimeter of a rectangle is 48 cm. Find the length of each side if the ratio
of their sides is 1 : 3?
Solution:
Since the ratio is 1 : 3, then we can represent their lengths in x and 3x.
x + x + 3x + 3x = 48
8x = 48
x = 6 cm., so the length of the other side is 3(6) = 18 cm.
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Ratio and Proportion //vvfernandez
Unit 3
RATIO AND PROPORTION
Applying the Extended Ratios
2. The angles of a rectangle has a ratio of 1 : 2 : 3 : 4. Find each angle, if the
sum of the angles of a rectangle is 3600.
Solution.
The ratio of their angles can be represented by x, 2x, 3x and 4x, thus
xo + 2xo + 3xo + 4xo = 360o
10xo = 360o
xo = 36
So, the angle measures are 36o, 2(36o) = 72o, 3(36o) = 108o, and 4(36o) = 144o
PRACTICE AND APPLICATIONS
A. Simplify the following ratios by making the units alike.
12 𝑖𝑛.
20 𝑘𝑔.
a. 20 𝑓𝑡.
b. 300 𝑔.
B. Use the number line to find the ratio of distances.
1.
4.
𝐴𝐶
𝐸𝐻
𝐷𝐹
𝐵𝐷
A
B
C D E F G H
I
1
2
3 4
9
5
𝐵𝐷
6 7
8
= _____
2. 𝐶𝐻 = _____
= ______
5.
3.
𝐴𝐹
𝐵𝐼
= _____
𝐴𝐺
𝐶𝐼
C. Word Problems.
1. The area of a rectangle is 100 cm2. The ratio of the width to its length is
1 : 4. Find the length and the width.
2. The measure of the angles in a triangle are in the extended ratio of
1 : 4 : 7. Find the measures of the angles.
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Ratio and Proportion //vvfernandez
Unit 3
RATIO AND PROPORTION
About the Proportion
On the other hand, when two ratios are equal then it is called proportion.
Properties of Ratio and Proportion
1. Reciprocal Property. If two ratios are equal, then their reciprocals are
also equal.
𝒂
𝒄
𝒃
𝒅
If
= , then =
𝒃
𝒅
𝒂
𝒄
2. Cross Product Property. The product of the means is equal to the
product of the extremes.
If
𝒂
𝒃
𝒄
= 𝒅, or a : b = c : d, then 𝒃𝒄 = 𝒂𝒅 𝒐𝒓 𝒂𝒅 = 𝒃𝒄
where (b and c) are the means & (a and d) are the extremes.
Here are the practical procedures in solving the ratio and proportion.
Example
If your car can go 350 miles on 20 gallons of gas, then at that rate
how much gas would you have to purchase to take a cross country
trip that was 3000 miles long?
Step 1. Establish a ratio of the given units.
Miles : Gas
Step 2. Provide a detailed proportion of the given units and let x to be the unknown
variable.
Miles : Gas = Miles : Gas
350 : 20 = 3000 : x
Step 3. Apply the cross product property to solve for x.
350
3000
=
20
𝑥
350x = 20(3000)
350x = 60,000
x = 171. 43 gallons of gas
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Ratio and Proportion //vvfernandez
Unit 3
RATIO AND PROPORTION
Solving the Extended Proportion
When three or more ratios are equal, thus it is called an extended proportion
Solve for the value of y and z.
6 : y : z = 10 : 30 : 50
Solution:
In fraction form,
6
10
=
𝑦
30
=
𝑧
50
Solving for y
Using the cross product
1. 6 : y = 10 : 30 ==> 10y = 6(30)
6
𝑦
=
10
30
==> 10y = 6(30)
10y = 180
y = 18
10y = 180
y = 18
Solving for z
2. 6 : z = 10 : 50 ==> 10z = 6(50)
10z = 300
z = 30
6
𝑧
=
10
50
==> 10z = 6(50)
10z = 300
z = 30
Thus, the resulting ratio is 6 : 18 : 30 which is equivalent to 10 : 30 : 50.
Checking the proportion and reducing the ratios to its lowest terms.
6
10
=
18
30
=
30
50
==>
3
5
=
3
5
=
3
5
When solving the extended proportion, express the ratios in fractional form and
select which one contains only one unknown/variable and solve using cross product
property.
Summary of Terms
a. Ratio – is a comparison of the two quantities, denoted by “ a : b” or “a/b”,
where b ≠ 0.
b. Proportion - is the equality of two ratios, such that “ a : b = c : d” or “a/b = c/d”.
c. Extended Proportion - is the equality of three or more proportions, such that
𝑎
𝑏
𝑐
“a : b : c = x : y : z” or 𝑥 = 𝑦 = 𝑧 .
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Ratio and Proportion //vvfernandez