1. In a 100 m race, A can give B 10 m and C 28 m. In the same race B can give C:
a) 18 m
b) 20 m
c) 27 m
d) 9 m
Explanation:
$$\eqalign{ & A:B = 100:90 \cr & A:C = 100:72 \cr & B:C = {B \over A} \times {A \over C} = {{90} \over {100}} \times {{100} \over {72}} = {{90} \over {72}} \cr} $$
When B runs 90 m, C runs 72 m.
When B runs 100m, C run
$$\left( {{{72} \over {90}} \times 100} \right)m = 80\,m$$
B can give C 20 m
2.A and B take part in 100 m race. A runs at 5 kmph. A gives B a start of 8 m and still beats him by 8 seconds. The speed of B is:
a) 5.15 kmph
b) 4.14 kmph
c) 4.25 kmph
d) 4.4 kmph
Explanation:
$$\eqalign{ & {\text{A's}}{\kern 1pt} {\kern 1pt} {\text{speed}} = \left( {5 \times \frac{5}{{18}}} \right){\text{m/sec}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{25}}{{18}}{\text{m/sec}} \cr & {\text{Time}}{\kern 1pt} {\kern 1pt} {\text{taken}}{\kern 1pt} {\kern 1pt} {\text{by}}{\kern 1pt} {\kern 1pt} {\text{A}}{\kern 1pt} {\kern 1pt} {\text{to}}{\kern 1pt} {\kern 1pt} {\text{cover 100 m}} \cr & = \left( {100 \times \frac{{18}}{{25}}} \right)\sec \cr & = 72\sec \cr & {\text{Time}}{\kern 1pt} {\kern 1pt} {\text{taken}}{\kern 1pt} {\kern 1pt} {\text{by}}{\kern 1pt} {\kern 1pt} {\text{B}}{\kern 1pt} {\kern 1pt} {\text{to}}{\kern 1pt} {\kern 1pt} {\text{cover 92 m}} \cr & = \left( {72 + 8} \right) = 80\sec \cr & {\text{B's}}{\kern 1pt} {\kern 1pt} {\text{speed}} = \left( {\frac{{92}}{{80}} \times \frac{{18}}{5}} \right){\text{kmph}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 4.14{\text{ kmph}} \cr} $$
3. In a 500 m race, the ratio of the speeds of two contestants A and B is 3 : 4. A has a start of 140 m. Then, A wins by:
a) 60 m
b) 40 m
c) 20 m
d) 10 m
Explanation: To reach the winning post A will have to cover a distance of (500 - 140)m, i.e., 360 m.
While A covers 3 m, B covers 4 m.
While A covers 360 m, B covers $$\left( {\frac{4}{3} \times 360} \right)$$ m = 480 m.
Thus, when A reaches the winning post, B covers 480 m and therefore remains 20 m behind.
A wins by 20 m
4. In a 100 m race, A beats B by 10 m and C by 13 m. In a race of 180 m, B will beat C by:
a) 5.4
b) 4.5 m
c) 5 m
d) 6 m
Explanation:
$$\eqalign{ & A:B = 100:90 \cr & A:C = 100:87 \cr & \frac{B}{C} = \frac{B}{A} \times \frac{A}{C} \cr & \,\,\,\,\,\,\,\, = \frac{{90}}{{100}} \times \frac{{100}}{{87}} \cr & \,\,\,\,\,\,\,\, = \frac{{30}}{{29}} \cr} $$
When B runs180m, C runs $$ \left( {\frac{{29}}{{30}} \times 180} \right){\text{m}} = 174{\text{m}}$$
B beats C by (180 - 174) m = 6 m
5. At a game of billiards, A can give B 15 points in 60 and A can give C to 20 points in 60. How many points can B give C in a game of 90?
a) 30 point
b) 20 points
c) 10 points
d) 12 points
Explanation:
$$\eqalign{ & A:B = 60:45 \cr & A:C = 60:40 \cr & \frac{B}{C} = {\frac{B}{A} \times \frac{A}{C}} = {\frac{{45}}{{60}} \times \frac{{60}}{{40}}} \cr & = \frac{{45}}{{40}} = \frac{{90}}{{80}} = 90:80 \cr & {\text{B}}\,{\text{can}}\,{\text{give}}\,{\text{C}}\,{\text{10}}\,{\text{points}}\,{\text{in}}\,{\text{a}}\,{\text{game}}\,{\text{of}}\,{\text{90}} \cr} $$
6. In a 300 m race A beats B by 22.5 m or 6 seconds. B's time over the course is:
a) 86 sec
b) 80 sec
c) 76 sec
d) None of these
Explanation:
$$\eqalign{ & {\text{B runs}}\frac{{45}}{2}{\text{m in 6 sec}} \cr & {\text{B covers 300m in}} \cr & = \left( {6 \times \frac{2}{{45}} \times 300} \right)\sec \cr & = 80\,{\text{sec}} \cr} $$
7. A runs $$1\frac{2}{3}$$ times as fast as B. If A gives B a start of 80 m, how far must the winning post be so that A and B might reach it at the same time?
a) 200 m
b) 300 m
c) 270 m
d) 160 m
Explanation: Ratio of speeds of A and B $$ = \frac{5}{3}:1 = 5:3$$
Thus, in race of 5m, A gains 2m over B
2m are gained by A in a race of 5m
80m will be gained by A in a race of
$$\left( {\frac{5}{2} \times 80} \right){\text{m}} = 200{\text{ m}}$$
Winning post is 200 m away from the starting point.
8.In a 100 m race, A can beat B by 25 m and B can beat C by 4 m. In the same race, A can beat C by:
a) 21 m
b) 26 m
c) 28 m
d) 29 m
Explanation:
$$\eqalign{ & A:B = 100:75 \cr & B:C = 100:96 \cr & A:C = {\frac{A}{B} \times \frac{B}{C}} \cr & = {\frac{{100}}{{75}} \times \frac{{100}}{{96}}} \cr & = \frac{{100}}{{72}} \cr & 100:72 \cr & A\,\text{beats}\,C\,by \cr & = \left( {100 - 72} \right)m \cr & = 28\,m \cr} $$
9. In a race of 200 m, B can give a start of 10 m to A and C can give a start of 20 m to B. The start that C can give to A in the same race is ?
a) 27 m
b) 29 m
c) 30 m
d) 25 m
Explanation: B : A = 200 : 190
C : B = 200 : 180
$$\eqalign{ & = \frac{C}{A} \cr & = {\frac{C}{B} \times \frac{B}{A}} \cr & = {\frac{{200}}{{180}} \times \frac{{200}}{{190}}} \cr & = \frac{{200}}{{171}} \cr} $$
C can give to A, a start of (200 - 171) m = 29 m.
10. In a kilometre race, A beats B by 30 seconds and B beats C by 15 seconds. If A beats C by 180 m, the time taken by A to run 1 kilometre, is ?
a) 200 sec
b) 205 sec
c) 210 sec
d) 250 sec
Explanation: In a km race, suppose A takes t sec,
Then, B takes (t + 30) sec, and C takes (t + 45) sec
180 m is covered by C in 45 sec
1000 m is covered by C in
= $$\left( {\frac{{45}}{{180}} \times 1000} \right)$$ sec
= 250 sec
A covers 1000 m in
= (250 - 45) sec
= 205 sec