Question

7. Two numbers are in the ratio 4: 5. If 30 is
subtracted from each of the numbers, the ratio
becomes 1 : 2. Find the numbers.​

Answer 1

$\bf{\huge{\underline{\boxed{\rm{\red{Answer:}}}}}}$

$\bf{\underline{\underline{\sf{\blue{Given:}}}}}$

• Two numbers are in the ratio 4: 5
• If 30 is subtracted from each of the numbers, the ratio becomes 1:2

$\bf{\underline{\underline{\sf{\blue{Given:}}}}}$

• The numbers

$\bf{\underline{\underline{\sf{\blue{Solution:}}}}}$

Let x be the common multiple of the ratio 4:5

•°• Smaller number = 4x

Greater number = 5x

$\bf{\underline{\underline{\sf{\blue{As\:per\:the\:question}}}}}$

• If 30 is subtracted from each of the numbers, the ratio becomes 1:2

After subtracting 30 :-

Smaller number = 4x - 30

Greater number = 5x - 30

Ratio = 1:2

Representing the condition mathematically,

=> $\bf\large\frac{4x-30}{5x-30}$ = $\bf\large\frac{1}{2}$

Cross multiplying,

=> 2 (4x - 30) = 5x -30

=> 8x - 60 = 5x - 30

=> 8x - 5x = - 30 + 60

=> 3x = 30

=> x = $\large\frac{30}{3}$

=> x = 10

Substitute x = 10 in the values of the ratio,

$\bf{\large{\underline{\boxed{\rm{\pink{Smaller\:number=\:4x\:=\:4\times\:10\:=\:40}}}}}}$

$\bf{\large{\underline{\boxed{\rm{\pink{Greater\:number=\:5x\:=\:5\times\:10\:=\:50}}}}}}$

$\bf\huge\underline{Verification}$

For first case :-

• Two numbers are in the ratio 4: 5

Smaller number = 4x = 40

Greater number = 5x = 50

Ratio = 4:5

=> $\bf\large\frac{4x}{5x}$ = $\bf\large\frac{4}{5}$

=> $\bf\large\frac{40}{50}$ = $\bf\large\frac{4}{5}$

Dividing LHS by 10,

=> $\bf\large\frac{4}{5}$ = $\bf\large\frac{4}{5}$

LHS = RHS

For second case :-

• If 30 is subtracted from each of the numbers, the ratio becomes 1:2

Smaller number = 4x - 30 = 40 - 30 = 10

Greater number =5x - 30 = 50 - 30 = 20

Ratio = 1:2

=> $\bf\large\frac{4x-30}{5x-30}$ = $\bf\large\frac{1}{2}$

=> $\bf\large\frac{10}{20}$ = $\bf\large\frac{1}{2}$

Dividing LHS by 10,

$\bf\large\frac{1}{2}$ = $\bf\large\frac{1}{2}$

LHS = RHS.

Hence verified.

Answer 2

$\huge\sf{Answer:-}$

Let y be the common multiplies and let 4x be the samller number and let 5x be the greater number and also let 4x - 30 be the smaller number and 5x - 30 be the greater number this is for both the cases.

So,

4 x - 30 is the smaller number and 5x - 30 be the great number.

4x-30/5x-30 = 1/2

We can cross multiple

= 2(4x-30) = 5x - 30

= 8x - 60 = 5x - 30

= 8x - 5x = -30 + 60

= 3x = 30 - x =30/3 = x=10

Adding values x = 10 for ratios 4x = 4 × 10 = 40 is the smaller number.

5x = 5 ×10=50 is the greater number.

Case (1)

4:5 be the ratio of both two numbers.

4x=40

5x=50

4:5 is the Ratio

4x/5x = 4/5 = 40/50 = 4/5

taking LHS =10

4/5 = 4/5

So, LHS = RHS

Case (2)

1:2 is the ration if we subtract 30 from each number.

4x-30=40-30=10

5x-30=50-30=20

1:2 is the ratio

4x-30/5x-30 = 1/2 * 10/20 = 1/2

Taking LCM

1/2 = 1/2

So, LHS = RHS

Therefore,proved!!

Refer the attachments.