Question

Rohan, Taruna, and Sahil went for an evening walk. Rohan covers 75 cm in each step, while Taruna and Sahil cover 80 cm and 85 cm, respectively. If all three of them start together, what is the minimum distance traveled by the three, so that all of them cover an equal distance?

Answer 1

Hint: LCM determines the least multiple of the given numbers. Take LCM of the measure of one step of one person, use the factorization method to get the factors. After that multiply them to get the LCM of their steps. Then convert the distance from centimeters to meters. The value obtained is the desired result.

Complete step by step answer:
Given, Rohan covers 75 cm in each step, while Taruna and Sahil cover 80 cm and 85 cm, respectively.
To find: - The minimum distance each of them walk to cover an equal distance
As we know LCM stands for least common multiple i.e. LCM of the given numbers will be the least number which is divisible by given numbers. Now coming to the question, it is given that the length of the steps by three persons is 75 cm, 80 cm, and 85 cm.
So, they will meet at a certain distance which should be a multiple of given numbers because the length covered by each person will be multiple measures of one step of each person.
Thus, the distance at which all three persons will meet for the very first time is the LCM of the measures of one step of each person i.e. LCM of 75, 80, 85. So, we can find the LCM of 75, 80, 85 by doing prime factorization of them.
$ \Rightarrow 75 = 3 \times 5 \times 5$
$ \Rightarrow 80 = 2 \times 2 \times 2 \times 2 \times 5$
$ \Rightarrow 85 = 5 \times 17$
So, the L.C.M. of the steps is,
$ \Rightarrow $L.C.M. (75, 80, 85)$ = 2 \times 2 \times 2 \times 2 \times 3 \times 5 \times 5 \times 17$
Multiply the terms on the right side,
$ \Rightarrow $L.C.M. (75, 80, 85)$ = 20400$cm
Convert the distance into the meter,
$ \Rightarrow 20400{\text{cm}} = \dfrac{{20400}}{{100}}{\text{m}}$
Cancel out the common factors,
$ \Rightarrow 20400{\text{cm}} = 204{\text{m}}$

Hence, all three persons will meet at 204m distance for the first time, or in other words the minimum distance each should walk so that each can cover the same distance in complete steps is 204m.

Note:
The students might make mistakes, as one may calculate HCF of the lengths in place of LCM of them. It is common sense that HCF will be lower than the given numbers. So, it cannot represent the required distance. LCM can also be calculated by individual prime factorization of the numbers as well.
LCM = multiply the common numbers at once and hence with other numbers.