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P $8$ . The ratio of two numbers is $4:5$ after adding $20$ to the smaller number and Subtracting $20$ from the greater number the ratio becomes $14:13$ . Find the numbers.

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Write an equivalent ratio

Full solution

Q. P

8

. The ratio of two numbers is

4:5

after adding

20

to the smaller number and Subtracting

20

from the greater number the ratio becomes

14:13

. Find the numbers.

Identify Ratio Equations: Identify the initial ratio and set up the equations using variables. $\newline$ Let the smaller number be $4x$ and the larger number be $5x$ .
Set Up Equations: Apply the changes to the numbers and set up the second ratio equation. $\newline$ $(4x + 20) / (5x - 20) = \frac{14}{13}$
Cross-Multiply: Cross-multiply to solve for $x$ . $13(4x + 20) = 14(5x - 20)$ $52x + 260 = 70x - 280$
Solve for x: Simplify and solve for x. $\newline$ $70x - 52x = 260 + 280$ $\newline$ $18x = 540$ $\newline$ $x = 30$
Substitute Back: Substitute $x$ back into the original numbers. $\newline$ Smaller number = $4x = 4(30) = 120$ $\newline$ Larger number = $5x = 5(30) = 150$

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