Question

Two numbers are the ratio 5:6 if 8 is subtracted from each of the numbers the ratio becomes 4:5 find the number

Answer 1

Two numbers are in the ratio 5:6.

Let one number be 5M and other number be 6M.

If 8 is subtracted from each number then the ratio becomes 4:5.

Now,

New numbers -

• One number = 5M - 8
• Other number = 6M - 8

$\bold{According \:to\: question}$

=> $\sf{\dfrac{5M\:-\:8}{6M\:-\:8}}\:=\:\dfrac{4}{5}$

$\bold{Cross \:multiply \:them}$

=> $\sf{5(5M\:-\:8)\:=\:4(6M\:-\:8)}$

=> $\sf{25M\:-\:40\:=\:24M\:-\:32}$

=> $\sf{25M\:-\:24M\:=\:-32\:+\:40}$

=> $\sf{M\:=\:8}$

∴ One number = 5M

=> $\sf{5(8)}$

=> $\sf{40}$

∴ Other number = 6M

=> $\sf{6(8)}$

=> $\sf{48}$

•°• $\bold{\sf{\underline {Numbers\:are\:40\:and\:48}}}$

Answer 2

$\bold{\underline{\underline{Answer:}}}$

Smaller Number = 40

Greater Number = 48

$\bold{\underline{\underline{Step\:-\:by\:-\:step\:explanation:}}}$

Given :

• Two numbers are the ratio 5:6
• If 8 is subtracted from each of the numbers the ratio becomes 4:5

To find :

• The numbers

Solution :

Let x be the common multiple of the ratio 5:6

•°• Greater Number = 6x

Smaller Number = 5x

$\bold{\underline{\underline{As\:per\:question:}}}$

• 8 is subtracted from each of the numbers the ratio becomes 4:5

•°• Greater Number = 6x - 8

Smaller Number = 5x - 8

Constituting it mathematically,

$\implies$ $\bold{\frac{5x-8}{6x-8}}$ = $\bold{\frac{4}{5}}$

Cross multiplying,

$\implies$ $\bold{5(5x-8)\:=\:4(6x-8)}$

$\implies$$\bold{25x\:-40\:=\:24x\:\:-32}$

$\implies$ $\bold{25x\:-24x\:=\:-32\:+\:40}$

$\implies$ $\bold{x\:=8}$

•°• x = 8

Substitute x = 8 in the values of the ratio,

•°• Smaller number = 5x = 5 (8) = 40

Greater Number = 6x = 6(8) = 48

$\bold{\underline{Verification}}$

For first case :

• Two numbers are the ratio 5:6

Greater Number = 48

Smaller Number = 40

Ratio = 5:6

$\implies$ $\bold{\frac{40}{48}}$ = $\bold{\frac{5}{6}}$

Dividing LHS by 8,

$\implies$ $\bold{\frac{5}{6}}$ = $\bold{\frac{5}{6}}$

LHS = RHS.

For second case :

• 8 is subtracted from each of the numbers the ratio becomes 4:5

Greater Number = 48 - 8 = 40

Smaller Number = 40 - 8 = 32

Ratio = 4:5

$\implies$ $\bold{\frac{32}{40}}$ = $\bold{\frac{4}{5}}$

Dividing LHS by 8,

$\implies$ $\bold{\frac{4}{5}}$ = $\bold{\frac{4}{5}}$

LHS = RHS.

Hence Substantiated.