Question

the denominator of a rational number is greater than its numerator by 3. If the numerator is increased by 7 and the denominator is decreased by 1 the new number becomes 3/2 find the original number.
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Answer 1

Given :

• The denominator of a rational number is greater than its numerator by 3.
• If the numerator is increased by 7 and the denominator is decreased by 1, the new number becomes 3/2.

To find :

• Original number.

Solution :

Consider,

• Numerator = x
• Denominator = y

According to the 1st condition :-

• The denominator of a rational number is greater than its numerator by 3.

$\to\sf{y=x+3...............(i)}$

According to the 2nd condition :-

• If the numerator is increased by 7 and the denominator is decreased by 1, the new number becomes 3/2.

$\to\sf{\dfrac{x+7}{y-1}=\dfrac{3}{2}}$

$\to\sf{\dfrac{x+7}{x+3-1}=\dfrac{3}{2}\:[put\:y=x+3\: from\:eq(1)]}$

$\to\sf{\dfrac{x+7}{x+2}=\dfrac{3}{2}}$

$\to\sf{3x+6=2x+14}$

$\to\sf{3x-2x=14-6}$

$\to\sf{x=8}$

• Numerator = 8

Now put x = 8 in eq(1)

$\to\sf{y=x+3}$

$\to\sf{y=8+3}$

$\to\sf{y=11}$

• Denominator = 11

Therefore,

${\boxed{\large{\bold{Original\: number=\dfrac{8}{11}}}}}$

Answer 2

AnsWer :

8 / 11.

SolutioN :

Let,

• Numerator be N.
• Denominator be D.

☛ Case 1.

• The denominator of a rational number is greater than its numerator by 3.

→ D = N + 3. ____ ( 1 )

$\rule{200}2$

☛ Case 2.

• If the numerator is increased by 7 and the denominator is decreased by 1 the new number becomes 3/2.

→ N + 7 / D - 1 = 3 / 2.

→ 2( N + 7 ) = 3( D - 1 )

→ 2N + 14 = 3D - 3.

→ 2N - 3D = - 3 - 14

→ 2N - 3D = - 17.

→ - 3D = - 17 - 2N.

→ 3D = 17 + 2N.

→ D = 17 + 2N / 3. _____( 2 )

$\rule{200}2$

✎ Now, From Equation ( 1 ) and ( 2 )

→ 17 + 2N / 3 = N + 3.

→ 17 + 2N = 3N + 9.

→ 17 = N + 9.

→ N = 17 - 9.

→ N = 8.

✎ Now, Putting the value of N in Equation ( 1 ) We get.

→ D = N + 3.

→ D = 8 + 3.

→ D = 11.

Our Fraction become N / D → 8 / 11.

✡ Therefore, the value of Original number is 8 / 11.