GATE Civil 2022 Set 2 | GA | Question: 3

Question

Both the numerator and the denominator of $\dfrac{3}{4}$ are increased by a positive integer, $x,$ and those of $\dfrac{15}{17}$ are decreased by the same integer. This operation results in the same value for both the fractions.

What is the value of $x ?$

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

According to the question, $\frac{3+x}{4+x} = \frac{15-x}{17-x}$

$\Rightarrow (3+x)(17-x) = (4+x)(15-x)$

$\Rightarrow \require{cancel} 51-3x + 17x – {\color{Red}{\cancel{x^{2}}}} = 60 + 15x-4x – {\color{Red}{\cancel{x^{2}}}}$

$\Rightarrow 14x-11x = 60-51$

$\Rightarrow 3x = 9$

$\Rightarrow {\color{Blue}{\boxed{x = 3}}}$

Correct Answer $:\text{C}$

Comments

Fraction Properties:

1. Basic Equality: If \(\dfrac{a}{b} = \dfrac{c}{d}\), then:

• \(\dfrac{a-b}{b} = \dfrac{c-d}{d}\)
• \(\dfrac{a+b}{b} = \dfrac{c+d}{d}\)

2. Componendo and Dividendo: If \(\dfrac{a}{b} = \dfrac{c}{d}\), then:

• \(\dfrac{a}{a-b} = \dfrac{c}{c-d}\)
• \(\dfrac{b}{a-b} = \dfrac{d}{c-d}\)

3. Mediant Property: If \(\dfrac{a}{b} < \dfrac{c}{d}\), then:

•    \(\dfrac{a}{b} < \dfrac{a+c}{b+d} < \dfrac{c}{d}\)

4. Cross Product: If \(\dfrac{a}{b} = \dfrac{c}{d}\), then:

•    \(ad = bc\)

5. Proportion Generation: If \(\dfrac{a}{b} = \dfrac{c}{d}\), then:

•    \(\dfrac{ma+nc}{mb+nd} = \dfrac{a}{b} = \dfrac{c}{d}\) (for any $m$ and $n$)

6. Reciprocal Property: If \(\dfrac{a}{b} = \dfrac{c}{d}\), then:

•    \(\dfrac{b}{a} = \dfrac{d}{c}\)

7. Square Relationship: If \(\dfrac{a}{b} = \dfrac{c}{d}\), then:

• \(\dfrac{a^2+b^2}{ab} = \dfrac{c^2+d^2}{cd}\)

8. Subtraction Property: If \(\dfrac{a}{b} = \dfrac{c}{d}\), then:

•    \(\dfrac{a-c}{b-d} = \dfrac{a}{b} = \dfrac{c}{d}\) (if \(b \neq d\))

9. Alternation: If \(\dfrac{a}{b} = \dfrac{c}{d}\), then:

• \(\dfrac{a}{c} = \dfrac{b}{d}\)

10. Composition: If \(\dfrac{a}{b} = \dfrac{c}{d} = \dfrac{e}{f}\), then:

•     \(\dfrac{a+c+e}{b+d+f} = \dfrac{a}{b} = \dfrac{c}{d} = \dfrac{e}{f}\)

Where, a, b, c, and d are positive integers

PS: Using these properties we can easily solve this type of questions with ease, like this:

$\dfrac{3+x}{4+x} = \dfrac{15-x}{17-x}$

$\dfrac{3+x - (4+x)}{4+x} = \dfrac{15-x - (17-x)}{17-x}$

$\dfrac{3 + x - 4 - x}{4+x} = \dfrac{15 - x - 17+ x}{17-x}$

$\dfrac{-1}{4+x} = \dfrac{-2}{17-x}$

$1(17-x) = 2(4+x)$

$17-x = 8+2x$

$17-8 =3x$

$9 =3x$

$\boxed{x = 3}$

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bro one suggesation instead of writing answers as a comment u can post it as a answer if u are giving if some normal discussion then u can do in comments