Question

1. Two numbers are in the ratio 7: 11. If 7 is added to each of the numbers, the ratio becomes 2: 3. Find the numbers.

2. What number must be added to each term of the ratio 7 : 12 to make it 2 : 3​

Answer 1

Given: (1)

• Two numbers are in the ratio 7: 11. If 7 is added to each of the numbers, the ratio becomes 2: 3. Find the numbers.

Solution:

The given ratio is 7 : 11.

Let the two numbers be 7x and 11x, where

• x is the consonant of proportionality.

${ \boldsymbol{ \underline{ \pink{According \: to \: the \: question : }}}}$

$\\ \sf \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:\frac{7x + 7}{11x + 7} = \frac{2}{3}$

$\: \: \: \: \implies \sf \: 3(7x + 7) = 2(11x + 7) \\ \\ \: \: \: \: \implies \: \sf \: \: 21x + 21 = 22x + 14 \\ \\ \implies \sf \: 21x - 22x = 14 - 21 \\ \\ \: \: \: \: \implies \: \: \: \: \: \: \: \: \: \:\sf \: - x \: \: \: \: = - 7\: \: \: \: \: \: \: \: \: \: \: \\ \\ \implies \: \: \: \: \: \: \: \: \: \:\sf \: x \: \: \: \: = 7\: \: \: \: \: \: \: \: \: \: \: \: \:$

So,

• 7x = 7 × 7 = 49

• 11x = 11 × 7 = 77

⠀⠀⠀Hence, The numbers are 49 and 76

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Given: (2)

• What number must be added to each term of the ratio 7 : 12 to make it 2 : 3?

Solution:

The given ratio is 7 : 12.

Let the required number to be added to each term of the ratio be x .

$\\ \sf \: Then, \: \: \: \: \: \: \: \: \: \: \: \: \frac{7 + x}{12 + x} = \frac{2}{3}$

$\\ \sf \implies 3(7 + x) = 2(12 + x) \\ \\ \sf \implies \: 21 + 3x = 24 + 2x \\ \\ \sf \implies \: 3x - 2x = 24 - 21 \\ \\ \sf \implies \: \: \: \: \: \: \: \: \: \: \: \: \: x = 3 \: \: \: \: \: \: \: \: \: \: \:$

Hence, 3 must be added to each term of the ratio.

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⠀⠀⠀⠀⠀⠀Hope it helps you!

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