MATHS M.C.Q. [1 Marks Each]

1. 5

Solution:

Let the number that must be added to each term of the ratio 9 : 16 be x.

Then,

$(9+\text{x}):(16+\text{x})=2:3$

$\Rightarrow\frac{(9+\text{x})}{(16+\text{x})}=\frac{2}{3}$

$\Rightarrow3(9+\text{x})=2(16+\text{x})$

$\Rightarrow27+\text{3x}=32+\text{x}$

$\Rightarrow3\text{x}-2\text{x}=32-27$

$\therefore\text{x}=5$

So, 5 must be added to each term of the ratio 9 : 16 to make the ratio 2 : 3.

Hence, the correct alternative is option (a).

1. 3

Solution:

Let the least number that is to be subtracted from each term of the ratio 15 : 19 be x.

Then,

$(15-\text{x}):(19-\text{x})=3:4$

$\Rightarrow4(15-\text{x})=3(19-\text{x})$

$\Rightarrow60-\text{4x}=57-3\text{x}$

$\Rightarrow3\text{x}-4\text{x}=57-3\text{x}$

$\Rightarrow-\text{x}=-3$

$\therefore\text{x}=3$

So, 3 is the least number to be subtracted from each term of the ratio 15 : 19 to make the ratio 3 : 4.

Hence, the correct alternative is option (a).

1. 243

Solution:

Let the third proportional of 3 and 27 be x.

Then,

$3:27::27:\text{x}$

$\Rightarrow3:27=27:\text{x}$

$\Rightarrow\frac{3}{27}=\frac{27}{\text{x}}$

$\Rightarrow3\text{x}=27\times27$

$\Rightarrow\text{x}=\frac{27\times27}{3}$

$\Rightarrow\text{x}=27\times9$

$\therefore\text{x}=243$

So, the third proportional of 3 and 27 is 243.

Hence, the correct alternative is option (a).

4. 2 : 3

Solution:

As, $24:36=\frac{24}{36}=\frac23=2:3$

So, the simplest from of 24 : 36 = 2 : 3

Hence, the correct alternative is option (d).

1. 4

Solution:

Let the present ages of Renu and Ravi be 5x and 6x.

As, the sum of their present ages = 44 years

$\Rightarrow5\text{x}+6\text{x}=44$

$\Rightarrow11\text{x}=44$

$\Rightarrow\text{x}=\frac{44}{11}$

$\therefore\text{x}=4$

Now, the present age of Renu = 5 × 4 = 20 years and

the present ages of Ravi = 6 × 4 = 24 years

So, the difference of their ages = 24 - 20 = 4 years.

Hence, the correct alternative is option (a).

2. 25

Solution:

Since, the mean proportional of two positive numbers a and b is the positive number x such that $\frac{\text{a}}{\text{x}}=\frac{\text{x}}{\text{b}}$

$\Rightarrow\frac{\text{a}}{10}=\frac{10}{\text{b}}$

$\Rightarrow\text{ab}=100$

But $\text{a}=4\text{b}$

$\Rightarrow\text{4b}\times\text{b}=100$

$\Rightarrow\text{b}^2=\frac{100}{4}$

$\Rightarrow\text{b}^2=25$

$\Rightarrow\text{b}=\sqrt{25}$

$\Rightarrow\text{b}=5$

$\Rightarrow\text{a}=4\times5=20$

$\therefore\text{a}+\text{b}=20+5=25$

Hence, the correct alternative is option (b).

2. 12

Solution:

Let x be the mean proportional of 6 and 24.

Then,

$\frac{\text{6}}{\text{x}}=\frac{\text{x}}{24}$

$\Rightarrow\text{x}^2=6\times24$ (By cross multiplication)

$\Rightarrow\text{x}^2=144$

$\therefore\text{x}=12$

So, the mean proportional of 6 and 24 is 12.

Hence, the correct alternative is option (b).

2. 896

Solution:

Let the number of boys in the school be x.

Since, the ratio of boys and girls in the school = 9 : 5.

$\Rightarrow\frac{\text{Number of boys}}{\text{Number of girls}}=\frac{9}{5}$

$\Rightarrow\frac{\text{x}}{320}=\frac{9}{5}$

$\Rightarrow\text{5x}=320\times9$

$\Rightarrow\text{x}=\frac{320\times9}{5}$

$\Rightarrow64\times9$

$\Rightarrow\text{x}=576$

$\therefore$ The total strength of the school = 576 + 320 = 896

Hence, the correct alternative is option (b).

1. 18 years

Solution:

Let the present age of Ravish and Shikha be 3x and 8x, respectively,

After six years,

Age of Ravish = (3x + 6) years and

Age of Shikha = (8x + 6) years

Since, $(\text{3x}+6):(8\text{x}+6)=4:9$

$\Rightarrow\frac{\text{(3x}+6)}{\text{(8x}+6)}=\frac{4}{9}$

$\Rightarrow9(3\text{x}+6)=4(8\text{x}+6)$

$\Rightarrow27\text{x}+54=32\text{x}+24$

$\Rightarrow-5\text{x}=-30$

$\Rightarrow\text{x}=\frac{-30}{-5}$

$\Rightarrow\text{x}=6$

$\therefore\text{3x}=3\times6=18$

So, the present age of Ravish is 18 years.

Hence, the correct alternative is option (a).

3. 5 : 1

Solution:

Since,

$\frac{1}{12}:\frac{1}{60}$

$=\frac{1}{12}\div\frac{1}{60}$

$=\frac{1}{12}\times\frac{60}{1}$

$=\frac{60}{12}$

$=\frac{5}{1}$

$=5:1$

Hence, the correct alternative is option (c).

4. 6 : 4 : 3

Solution:

As, $\text{2a}=\text{3b}=4\text{c}$

$\Rightarrow\text{2a}=\text{3b}$ and $3\text{b}=\text{4c}$

$\Rightarrow\frac{\text{a}}{\text{b}}=\frac32$ and $\frac{\text{b}}{\text{c}}=\frac43$

$\Rightarrow\text{a}:\text{b}=6:4$ and $\text{b}:\text{c}=4:3$

$\therefore\text{a}:\text{b}:\text{c}=6:4:3$

Hence, the correct alternative is option (d).

2. 35

Solution:

Let the fourth term be x.

As, $3:5: :21:\text{x}$

$\Rightarrow\frac35=\frac{21}{\text{x}}$

$\Rightarrow3\text{x}=21\times5$

$\Rightarrow\text{x}=\frac{21\times5}{3}$

$\Rightarrow\text{x}=7\times5$

$\therefore\text{x}=35$

So, the fourth term is 35.

Hence, the correct alternative is option (b).

3. Rs. 360

Solution:

Let P's share be Rs. x.

Then Q's share = Rs. (840 - x)

As, P's share : Q's share = 3 : 4

$\Rightarrow\frac{\text{P's share}}{\text{Q's share}}=\frac34$

$\Rightarrow\frac{\text{x}}{(840-\text{x})}=\frac{3}{4}$

$\Rightarrow\text{4x}=3(840-\text{x})$

$\Rightarrow\text{4x}=3\times840-\text{3x}$

$\Rightarrow\text{4x}+\text{3x}=3\times840$

$\Rightarrow7\text{x}=3\times840$

$\text{x}=\frac{3\times840}{7}$

$\Rightarrow\text{x}=3\times120$

$\therefore\text{x}=360$

So, P's share is Rs. 360.

Hence, the correct alternative is option (c).

3. 8 : 3

Solution:

As, $\text{p}:\text{q}=2:5$

$\Rightarrow\frac{\text{p}}{\text{q}}=\frac{2}{5}$

Let $\text{p}=\text{2x}$ and $\text{q}=5\text{x}$

Now, $\frac{25\text{p}+14\text{q}}{5\text{p}+\text{7q}}$

$=\frac{25\times2\text{x}+14\times5\text{x}}{5\times2\text{x}+7\times5\text{x}}$

$=\frac{50\text{x}+70\text{x}}{10\text{x}+35\text{x}}$

$=\frac{120\text{x}}{45\text{x}}$

$=\frac83$

$=8:3$

Hence, the correct alternative is option (c).

1. $\frac7{11}$

Solution:

As, $\text{x}:\text{y}=1:1$

$\Rightarrow\frac{\text{x}}{\text{y}}=\frac{1}{1}$

$\Rightarrow\text{x}=\text{y}$

Now,

$\frac{\text{3x}+4\text{y}}{5\text{x}+6\text{y}}$

$=\frac{3\text{x}+4\text{x}}{5\text{x}+6\text{x}}$ (As, x = y)

$=\frac{7\text{x}}{11\text{x}}$

$=\frac{7}{11}$

Hence, the correct alternative is option (a).

1. $\frac{16}{13}$

Solution:

As, $\text{a}:\text{b}=2:5$

$\Rightarrow\frac{\text{a}}{\text{b}}=\frac{2}{5}$

Let $\text{a}=\text{2x}$ and $\text{b}=5\text{5x}.$

Then,

$\frac{3\text{a}+\text{2b}}{4\text{a}+\text{b}}$

$=\frac{3\times2\text{x}+2\times\text{5x}}{4\times2\text{x}+5\text{x}}$

$=\frac{6\text{x}+10\text{x}}{8\text{x}+5\text{x}}$

$=\frac{16\text{x}}{13\text{x}}$

$=\frac{16}{13}$

Hence, the correct alternative is option (a).

1. 2 : 3 : 4

Solution:

As, $\frac{\text{a}}{2}=\frac{\text{b}}{3}=\frac{\text{c}}{4}$

$\Rightarrow\frac{\text{a}}{2}=\frac{\text{b}}{3}$ and $\frac{\text{b}}{3}=\frac{\text{c}}{4}$

$\Rightarrow3\text{a}=2\text{b}$ and $4\text{b}=3\text{c}$ (By cross multiplication)

$\Rightarrow\frac{\text{a}}{\text{b}}=\frac{2}{3}$ and $\frac{\text{b}}{\text{c}}=\frac{3}{4}$

$\Rightarrow\text{a}:\text{b}=2:3$

$\therefore\text{a}:\text{b}:\text{c}=2:3:4$

Hence, the correct alternative is option (a).

2. 20 : 15 : 12

Solution:

As, $\frac{1}{\text{a}}:\frac{1}{\text{b}}:\frac{1}{\text{c}}=3:4:5$

$\Rightarrow\frac{1}{\text{a}}:\frac{1}{\text{b}}=3:4$ and $\frac{1}{\text{b}}:\frac{1}{\text{c}}=4:5$

$\Rightarrow\frac{1}{\text{a}}\div\frac{1}{\text{b}}=\frac{3}{4}$ and $\frac{1}{\text{b}}\div\frac{1}{\text{c}}=\frac45$

$\Rightarrow\frac{1}{\text{a}}\times\frac{\text{b}}{1}=\frac{3}{4}$ and $\frac{1}{\text{b}}\times\frac{\text{c}}{1}=\frac{4}{5}$

$\Rightarrow\frac{\text{b}}{\text{a}}=\frac{3}{4}$ and $\frac{\text{c}}{\text{b}}=\frac{4}{5}$

$\Rightarrow\frac{\text{a}}{\text{b}}=\frac{4}{3}$ and $\frac{\text{b}}{\text{c}}=\frac{5}{4}$ (Reciprocal of both sides)

$\Rightarrow\frac{\text{a}}{\text{b}}=\frac{4\times5}{3\times5}$ and $\frac{\text{b}}{\text{c}}=\frac{5\times3}{4\times3}$

$\Rightarrow\frac{\text{a}}{\text{b}}=\frac{20}{15}$ and $\frac{\text{b}}{c}=\frac{15}{12}$

$\Rightarrow\text{a}:\text{b}=20:15$ and $\text{b}:\text{c}=15:12$

$\therefore\text{a}:\text{b}:\text{c}=20:15:12$

Hence, the correct alternative is option (b).

2. 20 : 15 : 12

Solution:

As, $\text{a}:\text{b}=5:7$ and $\text{b}:\text{c}=6:11$

$\Rightarrow\frac{\text{a}}{\text{b}}=\frac57$ and $\frac{\text{b}}{\text{c}}=\frac{6}{11}$

$\Rightarrow\frac{\text{a}}{\text{b}}=\frac{5\times6}{7\times6}$ and $\frac{\text{b}}{\text{c}}=\frac{6\times7}{11\times7}$

$\Rightarrow\frac{\text{a}}{\text{b}}=\frac{30}{42}$ and $\frac{\text{b}}{\text{c}}=\frac{42}{77}$

$\Rightarrow\text{a}:\text{b}=30:42$ and $\text{b}:\text{c}=42:77$

$\therefore\text{a}:\text{b}:\text{c}=30:42:77$

Hence, the correct alternative is option (b).

2. 8 : 15

Solution:

As, $\text{a}:\text{b}=4:5$

$\Rightarrow\frac{\text{a}}{\text{b}}=\frac{4}{5}$

Also, $\text{b}:\text{c}=2:\text{3}$

$\Rightarrow\frac{\text{b}}{\text{c}}=\frac{2}{3}$

So, $\text{a}:\text{c}=\frac{\text{a}}{\text{c}}$

$=\frac{\text{ab}}{\text{bc}}$

$=\frac{\text{a}}{\text{b}}\times\frac{\text{b}}{\text{c}}$

$=\frac45\times\frac23$

$=\frac{8}{15}$

$=8:15$

Hence, the correct alternative is option (b).

3. 1 : 1

Solution:

As, a : b = 3 : 4

$\Rightarrow\frac{\text{a}}{\text{b}}=\frac34$

So, $\text{4a}:\text{3b}=\frac{4\text{a}}{3\text{b}}$

$=\frac43\times\frac{\text{a}}{\text{b}}$

$=\frac43\times\frac34$

$=\frac{12}{12}$

$=\frac11$

$=1:1$

Hence, the correct alternative is option (c).

3. $\frac{35}2$

Solution:

As, 8 : x :: 16 : 35

$\Rightarrow\frac{\text{8}}{\text{x}}=\frac{16}{35}$

$\Rightarrow\text{16x}=8\times35$ (By cross multiplication)

$\Rightarrow\text{x}=\frac{8\times35}{16}$ (Transposing 16 to R.H.S.)

$\therefore\text{x}=\frac{35}{2}$

Hence, the correct alternative is option (c).

4. 15 : 8

Solution:

As, $\text{2x}=3\text{y}$

$\Rightarrow\frac{\text{x}}{\text{y}}=\frac32$

And, $\text{4y}=\text{5z}$

$\Rightarrow\frac{\text{y}}{\text{z}}=\frac54$

Now, $\text{x}:\text{z}=\frac{\text{x}}{\text{z}}$

$=\frac{\text{xy}}{\text{yz}}$

$=\frac{\text{x}}{\text{y}}\times\frac{\text{y}}{\text{z}}$

$=\frac32\times\frac54$

$=\frac{15}{8}$

$=15:8$

Hence, the correct alternative is option (d).

1. 6 : 15

Solution:

Since, $2:5=\frac25=\frac{2\times3}{5\times3}=\frac{6}{15}=6:15$

So, the ratio equivalent to 2 : 5 is 6 : 15.

Hence, the correct alternative is option (a).