Real Numbers | Exercise 1.1

Question 7

There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point?

Solution

Understand the Question

Sonia comes back to the starting point every 18 minutes, and Ravi every 12 minutes.
They will meet at the starting point only at a time that is a common multiple of both 18 and 12.
The first time they meet again is the smallest common multiple, which is the Least Common Multiple (LCM) of 18 and 12.

\begin{array}{lccccc} \text{\textbf{Sonia}} & \boxed{18} & \boxed{\mathbf{36}} & \boxed{54} & \boxed{72} & \dots \\[0.5em] \text{\textbf{Ravi}} & \boxed{12} & \boxed{24} & \boxed{\mathbf{36}} & \boxed{48} & \dots \end{array}

First common time $\rightarrow$ 36 minutes

Step 1 · Prime factorise 18

Time taken by Sonia $= 18 \text{ minutes}$ . Diagram 1

\begin{aligned} 18 &= 2 \times 9 \\ &= 2 \times 3 \times 3 \\ &= 2^1 \times 3^2 \end{aligned}

Step 2 · Prime factorise 12

Time taken by Ravi $= 12 \text{ minutes}$ .

\begin{aligned} 12 &= 2 \times 6 \\ &= 2 \times 2 \times 3 \\ &= 2^2 \times 3^1 \end{aligned}

Step 3 · Calculate LCM

To find the LCM, take the product of the highest power of each prime factor involved:

Highest power of $2 = 2^2$
Highest power of $3 = 3^2$

\begin{aligned} \text{LCM}(12, 18) &= 2^2 \times 3^2 \\ &= 4 \times 9 \\ &= 36 \end{aligned}

Answer

36 minutes

Common Mistakes

Confusing HCF and LCM: Finding HCF gives $6$ minutes, which is less than a single full round. For recurring or meeting events, always compute LCM.
Power Selection Error: Choosing the lowest powers of prime factors instead of the highest powers when calculating the LCM.

Question 7

More questions in Exercise 1.1