Question

The ratio of the numerator and the denominator of a rational number is 3:4. if 4 is added to the numerator and 6 is subtracted from the denominator, then the rational number becomes 8/5. Then the original rational number is?

Please tell with full working

Answer 1

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✺Answer:

♦️GiveN

• Ratio of numerator and denominator = 3:4
• 4 is added to numerator and 6 is subtracted from denominator.
• New ratio of numerator and denominator = 8:5

♦️To FinD

• Find the original rational number.

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✺Explanation of Q.

The above question is asking about a fraction in which the ratio of numerator to denominator is 3:4, when 4 is added to numerator and 6 is subtracted from denominator, the fraction changes and now the ratio becomes 8:5. So, we have to find the numerator and denominator of original fraction and the original fraction itself.

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✺Concept to be used:

The question can be done using two methods.

Method-1 : By forming a linear equation in one variable and solving it.

Method-2 : By taking 2 variables for numerator and denominator separately and solving it accordingly.

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✺Solution:

$\large{ \rm{ \underline{ \underline{ \pink{Method - 1}}}}}$

According to question,

Ratio of Numerator to Denominator = 3:4

So, let the Numerator and denominator be 3x and 4x

Now, 4 is added to numerator, New Numerator =3x+4

6 is subtracted from denominator, New Denominator = 4x-6

¤ New ratio = 8 : 5

So, this would be,

$\large{ \rm{ \rightarrow \: \frac{3x + 4}{4x - 6} = \frac{8}{5} }} \\ \\ \rm{\dag{ \green {cross \: multiplying} }} \\ \\ \large{ \rm{ \rightarrow \: 5(3x + 4) = 8(4x - 6)}} \\ \\ \large{ \rm{ \rightarrow \: 15x + 20 = 32x - 48}} \\ \\ \large{ \rm{ \rightarrow 15x - 32x = - 48 - 20}} \\ \\ \large{ \rm{ \rightarrow \: \cancel{- }17x = \cancel{-} 68}} \\ \\ \large{ \rm{ \rightarrow \: x = \frac{68}{17} = 4}}$

♠️ Then, Numerator = 3x = 3(4) = 12

And Denominator = 4x =4(4) = 16

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$\large{ \rm{ \underline{ \underline{ \pink{Method - 2}}}}}$

Let the numerator be x and denominator be y

Then,

$\large{ \rm{ \rightarrow \: \frac{x}{y} = \frac{3}{4} }} \\ \\ \rm{\dag{ \green{cross \: multiplying}}} \\ \\ \large{ \rm{ \rightarrow 4x = 3y}} \\ \\ \large{ \rm{ \rightarrow x = \frac{3y}{4}..............(1)}}$

Now, 4 is added to numerator = x + 4

And 6 is subtracted from denominator = y-6

¤ New ratio = 8:5

Then,

$\large{ \rm{ \rightarrow \: \frac{x + 4}{y - 6} = \frac{8}{5}}} \\ \\ \rm{\dag{ \green{cross \: multiplying}}} \\ \\ \large{ \rm{ \rightarrow 5(x + 4) = 8(y - 6)}} \\ \\ \large{ \rm{ \rightarrow 5x + 20 = 8y - 48}} \\ \\ \large{ \rm{ \rightarrow \: 5x - 8y = - 68...............(2)}}$

Now, putting equation (1) in equation (2)

$\large{ \rm{ \rightarrow \: 5 \times \frac{3y}{4} - 8y = - 68}} \\ \\ \large{ \rm{ \rightarrow \frac{15y - 32y}{4} = - 68}} \\ \\ \large{ \rm{ \rightarrow \frac{ - 17y}{4} = - 68}} \\ \\ \large{ \rm{ \rightarrow y = \frac{ \cancel{ - 68} \: \: 4 \times 4}{ \cancel{ - 17}}}} \\ \\ \large{ \rightarrow \boxed{ \rm{ y = 16}} } \\ \\ \large{ \rightarrow \boxed{ \rm{x = \frac{3(16)}{4} = 12}}}$

♠️ Thus, Numerator is 12

And denominator is 16.

✒Hence, Solved!

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Answer 2

Answer:

The original rational number is : $\rm\red{\frac{3}{4}}$

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Given:

▪️The ratio of the numerator and the denominator of a rational number is 3:4.

▪️4 is added to the numerator and 6 is subtracted from the denominator, then the rational number becomes 8/5.

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To Find:

The original rational number

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Solution :

Let the original rational number be , $\rm{\frac{m}{n}}$

By first condition:

Ratio of rational numbers is : $\rm{\frac{3}{4}}$

i.e.

$\frac{m}{n} = \frac{3}{4}$

therefore,

$4m = 3n$

$\rm\therefore{ m = \frac{3n}{4} ...................1)}$

If 4 is added to numerator and 6 is subtracted from numerator then the rational number becomes 8/5

$\rm\therefore{ \frac{m+4}{n-6} = \frac{8}{5}}$

$\rm\therefore{ 5m+20 = 8n-48}$

$\rm\therefore{ 68 = 8n-5m }$

now , putting value of m from equation 1)

we get,

$\rm\implies{ 68 = 8m- \frac{3n}{4} }$

$\rm\implies{ 68 = 8m- 5( \frac{3n}{4} )}$

$\rm\implies{ 68 = 8m- ( \frac{15n}{4} )}$

$\rm\implies{ 68 = \frac{32n -15n}{4} )}$

..........(by taking LCM on right hand side )

$\rm\implies{68 \times 4 =17n}$

$\rm\implies{( \frac{68 \times 4}{17} ) =n}$

$\rm\implies{4 \ times 4= n}$

$\rm{\red{\boxed{\green{n=16}}}}$

put the value of n in equation 1)

we get,

$m = \frac{3n}{4} = \frac{3 \times 16}{4}$

$m = 4 \times 3 = 12$

$\rm{\red{\boxed{\green{m=12}}}}$

The original rational number is : $\rm\red{ \frac{m}{n} = \frac{12}{16} \frac{3}{4}}$