Ratio and Proportion 1/2

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9. If $18:x = x:8$, then $x$ is equal to :
a. 144
c. 72
c. 26
d. 12

Answer: D
Explanation:
$18:x = x:8$ can be written in the fraction format as $\dfrac{{18}}{x} = \dfrac{x}{8}$
$\therefore 18 \times 8 = {x^2}$
$\therefore x = \sqrt {144} = 12$

10. If $a:b=c:d$, then $\displaystyle\frac{{ma + nc}}{{mb + nd}}$ is equal to
a. m : n
b. na:mb
c. a : b
d. md:nc

Answer: C
Explanation:
$a:b=c:d$ can be written in the fraction format as $\dfrac{{a}}{b} = \dfrac{c}{d}$
Let $\displaystyle\frac{a}{b} = \displaystyle\frac{c}{d} = k$.
$ \therefore \dfrac{a}{b} = k \Rightarrow a = kb$
$ \therefore \dfrac{c}{d} = k \Rightarrow c = kd$
$\displaystyle\frac{{ma + nc}}{{mb + nd}} = \displaystyle\frac{{m(kb) + n(kd)}}{{mb + nd}} = k\left[ {\displaystyle\frac{{mb + nd}}{{mb + nd}}} \right]$ $= k$
But $k = \displaystyle\frac{a}{b}$ So the required ratio = $a : b$

11. Rs.1050 is divided among P, Q and R. The share of P is $\displaystyle\frac{2}{5}$ of the combined share of Q and R. Thus, P gets:
a. Rs.200
b. Rs.300
c. Rs.320
d. Rs.420

Answer: B
Explanation:
Given, $P = \dfrac{2}{5}\left( {Q + R} \right)$ $ \Rightarrow \left( {Q + R} \right) = \dfrac{5}{2}P$
But, $P + Q + R = 1050$
Substituting the value of $Q + R$,
$ \Rightarrow P + \dfrac{5}{2}P = 1050$
$ \Rightarrow \dfrac{7}{2}P = 1050$
$ \Rightarrow P = \dfrac{2}{7} \times 1050$ $\require{cancel}= \dfrac{2}{\cancel{7}} \times {\cancel{1050}^{150}}$ $=300$
$\therefore$ $P = 300$

Alternative Method:
Let $Q + R$ got $5x$ units then $P$ gets $2x$ units.
But total $P + Q + R = 7x$ units. So,
P's share = Rs.$\left( {1050 \times \displaystyle\frac{2x}{7x}} \right) = Rs.300$

12. If Rs.600 is divided among $A, B, C$ in the ratio 2 : 3 : 5, find the share of $C$
a. Rs.280
b. Rs.300
c. Rs.170
d. Rs.200

Answer: B
Explanation:
Formula: If Rs.$K$ is divided among $A, B, C...$ in the ratio $a, b, c...$, then
Share of $A$ $ = \dfrac{a}{{a + b + c...}} \times K$
Share of $B$ $ = \dfrac{b}{{a + b + c...}} \times K$
and so on.
Share of $C$ $ = \dfrac{5}{{2 + 3 + 5}} \times 600$ $ = \dfrac{5}{{10}} \times 600 = 300$

13. If Rs.600 is divided among $A, B, C$ in the ratio 2 : 3 : 5, find how much more $C$ will get than the share of $B$
a. Rs.280
b. Rs.300
c. Rs.170
d. Rs.200

Answer: A
Explanation:
There is no need to find individual shares.
The amount $C$ will get extra than $B$ $ = \dfrac{{5 - 3}}{{2 + 3 + 5}} \times 600$ $ = \dfrac{2}{{10}} \times 600$$ = 120$

14. Rs.1060 is divided into three parts in such a way that half of the first part, one-third of the second part and one-fifth of the third part are in the ratio 4 : 5 : 6. Find the second part.
a. Rs.600
b. Rs.400
c. Rs.300
d. Rs.180

Answer: C
Explanation:
Let the three parts $ = x, y, z$
$ \Rightarrow \dfrac{x}{2}:\dfrac{y}{3}:\dfrac{z}{5} = 4:5:6$
$ \Rightarrow x:y:z = 8:15:30$ ($\because $ equating respective terms)
Second part $ = \dfrac{{15}}{{8 + 15 + 30}} \times 1060$$\require{cancel} = \dfrac{{15}}{\cancel{53}} \times \cancel{1060}^{20}$$ = 300$

15. The incomes of A and B are in the ratio 5 : 4 and their expenses are in the ratio 7 : 3. If each saves Rs.1300, then find the income of A.
a. 144
c. 72
c. 26
d. 12

Answer: D
Explanation:
$\begin{array}{*{20}{|r|cc|}} \hline
{}&A&B \\[4px]\hline
{Income}&{5x}&{4x} \\[4px]
{Expenditure}&{7y}&{3y} \\[4px]\hline
{Savings}&{1300}&{1300} \\\hline
\end{array}$
We know that, Income - Expenditure = Savings.
$5x - 7y = 1300$ $\quad (1)$
$4x - 3y = 1300$ $\quad (2)$
Equating left most parts,
$5x - 7y = 4x - 3y$
$x = 4y$ $\quad (3)$
Substituting in equation $(1)$,
$\therefore$ $5\times 4y - 3y = 1300$
$\therefore$ $20y - 7y = 1300$
$\therefore$ $13y = 1300$
$\therefore$ $y = 100$
Putting it in equation $(3)$, $x = 400$
Income of $A$ $= 5\times 400 = 2000$

16. The ratio of money with Ram and Gopal is 7:17 and that with Gopal and Krishna is 7:17 . If Ram has Rs.490, Krishna has :
a. Rs.2890
b. Rs.2330
c. Rs.1190
d. Rs.2680

Answer: A
Explanation:
Ram: Gopal $= 7 : 17 = (7 : 17) \times 7 = 49 : 119$
Gopal : Krishna $= 7 : 17 = (7 : 17) \times 17 = 119 : 289$
Ram : Gopal : Krishna $= 49 : 119 : 289$
or Ram : Krishna = $49 : 289$
Thus, $49 : 289 = 490 : x$
x = $\displaystyle\frac{{289 \times 490}}{{49}} = 2890$

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Ratio and Proportion: FormulasRatio and Proportion: Exercise - 1Ratio and Proportion: Exercise - 2