# Elementary School Question on a race between a frog and a rabbit.

In a race, a frog tried to catch up with a rabbit which started 162m ahead of the frog. For every 3.8m jump that the frog made, the rabbit made a 0.8m jump at the same time. How many jumps would the frog have to make in order to catch up with the rabbit? I understand the way of solving this question would be,

$3.8-0.8=3$

To find the difference, after that, to find the amount of steps:

$162 ÷ 3 = 54 steps$

But I do not know how to explain my workings. My question is that how should I explain my workings to my younger brother so that he understands?

• In the first step you find the frog's relative speed (that one might be hard to explain to a younger kid). In the second you're dividing up the distance the frog needs to go into the distance he travels in each hop. For this you can start with smaller numbers and/ or draw pictures to help him understand. – user137731 Nov 23 '16 at 14:37
• Thanks all for the explanations! – bytk Nov 23 '16 at 14:43
• This is a simple integer linear program $\min \{ x\in \mathbb{Z} \mid (0.8 -3.8)\, x \le -162, x \ge 0 \}$. :-) – mvw Nov 23 '16 at 15:17

First, show him a couple of steps. Explain how

• After one jump, the rabit travelled $0.8$ while the frog travelled $3.8$
• After two jumps, the rabit travelled $1.6$ while the frog travelled $7.6$ so the difference is now $6$
• After $3$ jumps, the difference is $9$.

Now, we can see a pattern. We can see that after each jump, the difference between the two decreases by $3$.

So, now we simply need to know the answer to the question

How many times must the difference decrease by $3$ untill it will be $0$?

which, in other words, is the quesiton $3$ times what is $162$, and the answer is $162:3$.