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Two numbers are in the ratio 5:6. If 8 is subtracted from each of the numbers, the ratio becomes 4:5. Then the numbers are ____.

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Detailed Solution

Here we are given the ratios of two numbers as 5:6. Let us assume that these two numbers are x and y. Hence, the ratio of x and y is given as 5:6.
⇒

x : y ∷ 5 : 6

Now, we know that any ratio can be written in the fraction, i.e

x : y

can be written as

\frac{x}{y}

So,

x : y ∷ 5 : 6

can be written as

\frac{x}{y}

\frac{5}{6}

⇒

6 x = 5 y

⇒

6 x - 5 y = 0

.....(i)
Now we are given that when 8 is subtracted from each of the numbers then the ratio is 4:5. So, when 8 is subtracted, the numbers become

x - 8

and

y - 8

. And since the ratio of these numbers is 4:5. So,

x - 8 : y - 8 ∷ 4 : 5

Writing it in the form of the fraction, we get,
⇒

\frac{x - 8}{y - 8} = \frac{4}{5}

⇒

5 (x - 8) = 4 (y - 8)

⇒

5 x - 40 = 4 y - 32

⇒

5 x - 4 y = 8

……(ii)
Now, let us solve the equations (i) and (ii) using the substitution method to get the values of x and y. From (i), we have

6 x - 5 y = 0

⇒

6 x = 5 y

Dividing by 6 on both the sides, we get,
⇒

x = \frac{5 y}{6}

Putting this value of x in the equation (ii), we get,
⇒

5 (\frac{5}{6} y) - 4 y = 8

⇒

\frac{25}{6} y - 4 y = 8

Taking LCM of 6, we get,
⇒

\frac{25 y - 24 y}{6} = 8

⇒

\frac{y}{6} = 8

⇒y= 48
Putting in the value of y in equation (i), we get,
⇒

6 x - 5 (48) = 0

⇒

6 x

=240
⇒

x = 40

Hence the values of x and y are 40 and 48.
Therefore, the required numbers are 40 and 48.

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