In math proportions we will mainly learn about introduction or basic concepts of proportion and also about continued proportion.
Equality of two ratios is called a proportion.
We already learnt that —
Statement of equality of ratios is called proportion.
Let us consider the two ratios.
6 : 10 and 48 : 80
The ratio 6 : 10 in the simplest form can be written as 3 : 5 and the ratio 48 : 80 in the simplest form can be written as 3 : 5.
i.e., 6 : 10 = 48 : 80
So, we say that four numbers 6, 10, 48, 80 are in proportion and the numbers are called the terms of the proportion. The symbol used to denote proportion is :: .
We write 6 : 10 :: 48 : 80. It can be read as 6 is to 10 as 48 is to 80.
In general we know, if four quantities a, b, c, d are in proportion, then a : b = c : d
or, a/b = c/d or a × d = b ×c
Here,
• First and fourth terms (a and d) are called extreme terms.
• Second and third terms (b and c) are called mean terms.
• Product of extreme terms = Product of mean terms
• If a : b : : c : d, then d is called the fourth proportional of a, b, c.
Also,
• If a : b :: b : c, then we say that a, b, c are in continued proportion, then c is the third proportional of a and b.
• Also, b is called the mean proportional between a and C.
• In general if a, b, c are in continued proportion then b² = ac or b = √ac.
Workedout problems on proportions with the detailed explanation showing the stepbystep are discussed below to show how to solve proportions in different examples.
1. Determine if 8, 10, 12, 15 are in proportion.
Solution:
Product of extreme terms = 8 × 15 = 120
Product of mean terms = 10 × 12 = 120
Since, the product of means = product of extremes.
Therefore, 8, 10, 12, 15 are in proportion.
2. Check if 6, 12, 24 are in proportion.
Solution:
Product of first and third terms = 6 × 24 = 144
Square of the middle terms = (12)² = 12 × 12 = 144
Thus, 12² = 6 × 24
So, 6, 12, 24 are in proportion and 12 is called the mean proportional between 6 and 24.
3. Find the fourth Proportional to 12, 18, 20
Solution:
Let the fourth proportional to 12, 18, 20 be x.
Then, 12 : 18 :: 20 : x
⇒ 12 × x = 20 × 18 (Product of Extremes = Product of means)
⇒ x = (20 × 18)/12
⇒ x = 30
Hence, the fourth proportional to 12, 18, 20 is 30.
4. Find the third proportional to 15 and 30.
Solution:
Let the third proportional to 15 and 30 be x.
then 30 × 30 = 15 × x [b² = ac ]
⇒ x = (30 × 30)/15
⇒ x = 60
Therefore, the third proportional to 15 and 30 is 60.
5. The ratio of income to expenditure is 8 : 7. Find the savings if the expenditure is $21,000.
Solution:
Income/Expenditure = 8/7
Therefore, income = $ (8 × 21000)/7 = $24,000
Therefore, Savings = Income  Expenditure
= $(24000  21000) = 3000
6. Find the mean proportional between 4 and 9.
Solution:
Let the mean proportional between 4 and 9 be x.
Then, x × x = 4 × 9
⇒ x² = 36
⇒ x = √36
⇒ x = 6 × 6
⇒ x = 6
Therefore, the mean proportional between 4 and 9 is 6.
● Ratios and Proportions
● Ratios and Proportions  Worksheets
7th Grade Math Problems
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