Question

in a ratio of 5:6 2 is added to the first one and 3 is added to the second one the ratio becomes 4:5 find the ratio​

Answer 1

Given:

✰ Ratio of 2 number = 5:6

✰ If 2 is added to the first one and 3 is added to the second one the ratio becomes 4:5.

To Find:

✠ The numbers.

Solution:

Let first number be 5x and second number be 6x.

Adding 2 and 3 in first and second number respectively.

First Number = 5x + 2

Second Number = 6x + 3

According to the question,

➣ 5x + 2/6x + 3 = 4/5

➣ 5(5x + 2) = 4(6x + 3)

➣ 25x + 10 = 24x + 12

➣ 25x - 24x = 12 - 10

➣ x = 2

Therefore :

➤ First number = 5x

➤ First number = 5 × 2

➤ First number = 10

➤ Second number = 6x

➤ Second number = 6 × 2

➤ Second number = 12

∴ The both the numbers are 10 and 12 respectively.

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Answer 2

★ The two numbers are 10 and 12, respectively.

Step-by-step explanation

Analysis -

‎ ‎ ‎ ‎ ‎ ‎It has been given that two numbers are in the ratio 5:6. If 2 is added to the first number and 3 is added to the second number, the resulting ratio will be 4:5. We've been asked to find the two numbers.

Solution -

‎ ‎ ‎ ‎ ‎ ‎ In consonance with the question, let us consider the first and second number be 5y and 6y, respectively. Now, as per the analysis:

‎ ‎ ‎ ‎ ‎“Two is added to the first number”

‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎⇢ 5y + 2 (exp. 1)

‎“Three is added to the second number”

‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎⇢ 6y + 3 (exp. 2)

On writing expression 1 & 2 in fractional form, we get:

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \sf \pmb{ \dfrac{5y + 2}{6y + 3} }$

Now, according to the question, let us equate this fraction with 4:5,

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \sf \pmb{ \dfrac{5y + 2}{6y + 3} = \dfrac{4}{5} }$

Solving the equation

$\dashrightarrow \sf{ \dfrac{5y + 2}{6y + 3} = \dfrac{4}{5} } \\ \\ \dashrightarrow{ \sf{5(5y + 2) = 4(6y + 3)}} \\ \\ \dashrightarrow{ \sf{25y + 10 = 24y + 12}} \\ \\ \dashrightarrow{ \sf{25y - 24y = 12 - 10}} \\ \\ \dashrightarrow{ \sf{(25 - 24)y = 2}} \\ \\ \dashrightarrow{ \sf{1y = 2}} \\ \\ \dashrightarrow{ \sf{y = \dfrac{2}{1} }} \\ \\ \dashrightarrow \underline{\boxed{ \tt{\pmb{y = \frak2}}}}$

The value of the variable 'y' is 2. Now if we substitute the value of 'y' in the expressions formed for both numbers, we get the required answer.

$\twoheadrightarrow$ First number [5y] = 5(2) = 10

$\twoheadrightarrow$ Second number [6y] = 6(2) = 12

Therefore, the first number is 10 and the second number is 12.

Verification -

$\dashrightarrow{ \sf{ \dfrac{5y}{6y} = 5:6}} \\ \\ \dashrightarrow{ \sf{ \dfrac{5 \cancel{(2)}}{6 \cancel{(2)}} = \dfrac{5}{6} }} \\ \\ \dashrightarrow{ \sf{ \dfrac{5}{6} = \dfrac{5}{6} }}$

L.H.S. is equal to R.H.S. Hence, our answer is correct!

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