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CAT 2019 Slot 1 — QA Question 25

Basics of TSD/ProportinalityEasy

The wheels of bicycles A and B have radii 30 cm and 40 cm, respectively. While traveling a certain distance, each wheel of A required 5000 more revolutions than each wheel of B. If bicycle B traveled this distance in 45 minutes, then its speed, in km per hour, was

Answer & solution

Correct answer: 16π

  • A

    14π

  • B

    18π

  • 16π

  • D

    12π

Solution

Easy

For a fixed distance, revolutions are inversely proportional to wheel circumference (hence radius). Use this ratio with the "5000 extra revolutions" gap to find each count, recover the distance, then convert to speed over 45 minutes.

Radii: rA=30r_A=30 cm, rB=40r_B=40 cm. Same distance D=2πr×ND=2\pi r\times N, so N∝1rN\propto \dfrac{1}{r}. Also NA=NB+5000N_A=N_B+5000, and B covers DD in 4545 min =34=\tfrac34 h.

1

Revolution ratio. Since N∝1/rN\propto 1/r:

NANB=rBrA=4030=43(distance equal)\begin{aligned} &\frac{N_A}{N_B}=\frac{r_B}{r_A}=\frac{40}{30}=\frac{4}{3}\quad\text{(distance equal)} \end{aligned}
2

Solve with the gap. Use NA=NB+5000N_A=N_B+5000 in the ratio.

NB+5000NB=43(from step 1 and the gap)⇒ 3(NB+5000)=4NB⇒ NB=15000,NA=20000\begin{aligned} &\frac{N_B+5000}{N_B}=\frac{4}{3}\quad\text{(from step 1 and the gap)}\\ &\Rightarrow\ 3(N_B+5000)=4N_B\\ &\Rightarrow\ N_B=15000,\qquad N_A=20000 \end{aligned}
3

Find the distance. Use wheel A: D=2πrANAD=2\pi r_A N_A, with rA=30r_A=30 cm =30×10−5=30\times10^{-5} km.

D=2π(30×10−5)(20000) km(cm→km)⇒ D=12π km\begin{aligned} &D=2\pi(30\times10^{-5})(20000)\ \text{km}\quad\text{(cm}\to\text{km)}\\ &\Rightarrow\ D=12\pi\ \text{km} \end{aligned}
4

Speed of B. B covers DD in 34\tfrac34 h.

vB=D 3/4 =12π3/4=12π×43(from step 3)⇒ vB=16π km/h\begin{aligned} &v_B=\frac{D}{\,3/4\,}=\frac{12\pi}{3/4}=12\pi\times\frac{4}{3}\quad\text{(from step 3)}\\ &\Rightarrow\ v_B=16\pi\ \text{km/h} \end{aligned}
vB=16π km/hv_B=16\pi\ \text{km/h}

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CAT 2019 Slot 1 QA Q25: The wheels of bicycles A and B have radii 30 cm and 40 cm, respectively. While traveling a certain distance, e — Solution | TheCATExam