Question paper, Set 10

$3$ years $= 3 \times 12$ months $= 36$ months

$\displaystyle \frac{8 \text{ months}}{36 \text{ months}} = \frac{8}{36} = \frac{2}{9}$

Ratio $= 2 : 9$

Answer: $2 : 9$

Sum of the terms of the ratio $= 5 + 6 + 7 = 18$

$\displaystyle \text{1st share} = \frac{5}{18} \times 6300 = 1750$

$\displaystyle \text{2nd share} = \frac{6}{18} \times 6300 = 2100$

$\displaystyle \text{3rd share} = \frac{7}{18} \times 6300 = 2450$

Answer: ₹1750, ₹2100, ₹2450

Let the fourth proportional be $x$. Then $9 : 15 :: 12 : x$.

$9 \times x = 15 \times 12$

$\displaystyle x = \frac{15 \times 12}{9} = 20$

Answer: $20$

$\text{Mean proportional} = \sqrt{16 \times 81} = \sqrt{1296} = 36$

Answer: $36$

If $\displaystyle \frac{a}{b} = \frac{c}{d}$ then $\displaystyle \frac{a}{b} = \frac{c}{d} = \frac{al + cm}{bl + dm}$.

Here $l = 3$, $m = 4$, so $\displaystyle \frac{3a + 4c}{3b + 4d} = \frac{a}{b} = \frac{2}{5}$

Answer: $\displaystyle \frac{2}{5}$

Times enlarged | Length (cm)

$20{,}000$ | $4$

$1$ | $y$

Direct proportion: $x_1 y_2 = x_2 y_1$

$\displaystyle 20{,}000 \times y = 1 \times 4 \Rightarrow y = \frac{1 \times 4}{20{,}000} = \frac{1}{5000}$

$\displaystyle \frac{1}{5000}$ cm $= 0.0002$ cm $= 2 \times 10^{-4}$ cm

Answer: $\displaystyle \frac{1}{5000}$ cm

Work left $=$ work of 36 men for $25 - 10 = 15$ days

Men now $= 36 - 9 = 27$

Men and days are in inverse proportion: $x_1 y_1 = x_2 y_2$

$\displaystyle 36 \times 15 = 27 \times y \Rightarrow y = \frac{540}{27} = 20$

Answer: $20$ days

$\displaystyle 45\% = \frac{45}{100} = \frac{9}{20}$

Answer: $\displaystyle \frac{9}{20}$

$\displaystyle 2.5\% \text{ of } 640 = \frac{2.5}{100} \times 640 = 16$

Answer: $16$

Final value after an increase of $25\%$ $\displaystyle = \left(\frac{100 + R}{100}\right)P$

$\displaystyle = \frac{100 + 25}{100} \times 480 = \frac{125}{100} \times 480 = 600$

Answer: $600$

Population $n$ years ago $\displaystyle = \frac{P}{\left(1 + \frac{R}{100}\right)^n}$

$\displaystyle = \frac{21{,}632}{\left(1 + \frac{4}{100}\right)^2} = \frac{21{,}632}{\left(\frac{26}{25}\right)^2}$

$\displaystyle = 21{,}632 \times \frac{25}{26} \times \frac{25}{26} = 20{,}000$

Answer: $20{,}000$

Let the total sales be ₹$x$.

$\displaystyle 4\% \text{ of } x = 180 \Rightarrow \frac{4}{100}x = 180$

$\displaystyle x = \frac{180 \times 100}{4} = 4500$

Answer: ₹4500

$\text{Discount} = 25\% \text{ of } 960 = 240$

$\text{S.P} = 960 - 240 = 720$

$\displaystyle \text{C.P} = \frac{100}{100 - \text{Loss\%}} \times \text{S.P}$

$\displaystyle = \frac{100}{90} \times 720 = 800$

$\text{Loss} = 800 - 720 = 80$

Answer: Discount ₹240; S.P ₹720; C.P ₹800; loss ₹80

Let C.P of each $= x$ and S.P of each $= y$.

$\displaystyle 10x = 8y \Rightarrow \frac{y}{x} = \frac{10}{8} = \frac{5}{4}$

$y > x$, so there is a gain.

$\displaystyle \text{Gain \%} = \left(\frac{y}{x} - 1\right) \times 100 = 25\%$

Answer: Gain of $25\%$

$x_1 + x_2 + \cdots + x_n = n\bar{x}$

$\text{Sum} = 25 \times 36.4 = 910$

Answer: $910$

4 hours → 168 km

$\displaystyle 7 \text{ hours} \to \frac{7}{4} \times 168 = 294 \text{ km}$

Answer: $294$ km

$\displaystyle s_1 = \frac{d}{T_1}$, $\displaystyle s_2 = \frac{d}{T_2} \Rightarrow \frac{s_1}{s_2} = \frac{T_2}{T_1}$

$T_1 : T_2 = s_2 : s_1 = 56 : 35 = 8 : 5$

Answer: $8 : 5$

$\displaystyle \text{Speed} = 90 \times \frac{5}{18} = 25 \text{ m/s}$

To cross a post, the train travels its own length.

$\text{Length} = S \times T = 25 \times 12 = 300 \text{ m}$

Answer: $300$ m

$\displaystyle \text{Interest for 1st year} = 6000 \times \frac{10}{100} = 600$

(i) Simple interest: the 2nd year's interest is also $600$, so S.I $= 600 + 600 = 1200$

(ii) Compound interest: new principal $= 6000 + 600 = 6600$

$\displaystyle \text{Interest for 2nd year} = 6600 \times \frac{10}{100} = 660$

$\text{C.I} = 600 + 660 = 1260$

Answer: (i) ₹1200 (ii) ₹1260

$\displaystyle A = P\left(1 + \frac{R}{100}\right)^n$

$\displaystyle 7225 = 6400\left(1 + \frac{R}{100}\right)^2$

$\displaystyle \Rightarrow \left(1 + \frac{R}{100}\right)^2 = \frac{7225}{6400} = \frac{289}{256} = \left(\frac{17}{16}\right)^2$

$\displaystyle \Rightarrow 1 + \frac{R}{100} = \frac{17}{16} \Rightarrow \frac{R}{100} = \frac{1}{16}$

$\Rightarrow R = 6.25\%$

Answer: $6.25\%$ per annum