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.
Worked-out problems on proportions with the detailed explanation showing the step-by-step 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.
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7th Grade Math Problems
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