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Unit: Ratio & Proportion
Chapter: Ratio, Proportions & Unitary Method
Reference: – Introduction to Ratio, Comparison by Division, Simplest Form of Ratio, Equivalent Ratios, Comparison of Ratios, Proportion (Equality of Ratios), Direct and Inverse Proportion, Unitary Method (Finding Value of One Unit), Applications (Best Buys, Speed, Time and Work)
After studying this chapter, you should be able to understand:
- Introduction to Ratio & Comparison by Division
- Simplest Form and Equivalent Ratios
- Comparison of Ratios
- Proportion & Cross-Product Rule
- Direct and Inverse Proportion
- Unitary Method & Its Two Variations
- Real-Life Applications of Ratio, Proportion & Unitary Method
Ratio and Proportion
Ratio and proportion are the basic topics which are used in our day to day life. For example, while cooking we take the ingredients in a certain proportion or while contributing the money for picnic and many more.
Ratio
A ratio is said to be a comparison between two or more amounts of the same kind through division. As it is a ratio, it does not have any unit. The ratio is denoted by ” : ” sign.
Suppose, x and y are two quantities of same units, then the fraction is x/y and it is called a ratio of x to y. It is written as x:y . The x and y are called terms of the ratio. Here x is first term and y is second term. The order of the terms is important as the value can be reversed.
Simplest Form
The ratio is always expressed in its simplest form.
For example 1: Simplify the ratio of each of the following in simplest form:
- 4 score to 3 dozen
- 1.4 km to 350 m
Solution:
i. 1 score = 20 units
4 scores = 4 × 20 = 80 units
1 dozen = 12 units
3 dozen = 3 × 12 = 36 unit
Ratio of 4 score to 3 dozen = 20 : 9.
ii. 1 km = 1000 m
1.4Km = 1.4 × 1000 m = 1400 m
Ratio of 1.4 km to 350 m = 4:1.
Example 2: Simplify the following ratio:
- 223
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: 2.4
- 13 :1 12
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Solution:
i. Ratio = 223
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: 2.4 = 83 : 2410= 832410
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= 83 × 1024= 109
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= 10: 9.
ii Ratio = 13 :1 12
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= 13 : 32
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= 1332
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= 13 × 23= 29
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= 2: 9
Alternative Method: Ratio = 13 :1 12
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= 13 : 32
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We can take the L.C.M. of the ratio, which is 6
Then we will multiply the L.C.M. to each term
13 ×6=2
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: 32 ×6=9
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The new ratio = 29
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= 2: 9.
To find the greater ratio
It is easy to calculate which ratio is greater by the usage of L.C.M. and making the ratio in like fractions.
For example: 1. which ratio is greater 312 : 313
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or 4.8: 3.6?
Solution:
312 : 313
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= 72 : 103= 72 × 310=2120
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= 21: 20
or
4.8: 3.6 = 4.83.6= 4836= 43
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= 4: 3
L.C.M. OF 20 AND 3 IS 60
2120
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= 21×320×3
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= 6360
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and 43= 4×203×20= 8060
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As 80 >
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63, 8060 > 6360 →43 > 2120
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.
Hence, 4.8: 3.6 is greater ratio.
Hence, 4.8: 3.6 is greater ratio.
Missing Numbers
Ratio helps in finding out the missing numbers easily as there is a pattern to the ratio. With the help of multiples, we can find out the numbers. Let us look at the examples.
For example: Find the missing numbers.
2114= 3= 6
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. Find the missing numbers.
Solution:
Let us look at the pattern, in the numerator there is 21 and 3. We know that 3 × 7 = 21.
Accordingly, 14 ÷ 7 will be the answer for the second fraction, which will be 2.
The fractions are 2114= 32= 6
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.
In order to find the numerator of the third fraction, we will look at that pattern of the denominator.
In the denominator there 2 × 3 = 6.
The numerator should also be multiplied by the same number, i.e., 3 × 3 = 9.
Hence, the fraction is2114= 32= 96
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.
Distribution of certain unit in a ratio.
We always distribute a whole amount among people in ratio. Suppose there are three brothers and they have decided to distribute the property of their father among themselves equally. So in this case the ratio to each brother will be.
For example: 1. Divide $15 between Peter and Ron in the ratio 2:3.
Solution: Ratio of Peter share to Ron share = 2:3
Sum of the ratio terms = 2+3= 5
Therefore , Peter share = $ (2*15)/5 = $6
And , Ron share = $ (3*15 )/5 = $9.
