 # Problem of the Week Problem A and Solution Dog Walking

## Problem

Petra walks his dog once a day. Most days when Petra walks his dog, he takes a route that is $$3 \frac{1}{2}$$ km long. When it is raining, he does a shorter walk which is only $$2$$ km long.

One week it rained for $$3$$ days and did not rain on the other $$4$$ days. How far did Petra walk his dog that week? ## Solution

On each of the $$3$$ days it rained, Petra walked $$2$$ km for a total of $$2 + 2 + 2 = 6$$ km.

On each of the $$4$$ days it did not rain, Petra walked $$3 \frac{1}{2}$$ km.
We know that $$3 \frac{1}{2}$$ is the same as $$3 + \frac{1}{2}$$, so over four days, the total distance Petra walked is equal to $$3 + \frac{1}{2} + 3 + \frac{1}{2} + 3 + \frac{1}{2} + 3 + \frac{1}{2}$$.

Collecting the whole numbers and the fractions, we can rewrite this as $$3 + 3 + 3 + 3 + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \frac{1}{2}$$.
Since $$\frac{1}{2} + \frac{1}{2} = 1$$, we can rewrite this as $$3 + 3 + 3 + 3 + 1 + 1 = 14$$ km.

Alternatively, to calculate the distance Petra walked on the days it did not rain, we can add $$3 \frac{1}{2} + 3 \frac{1}{2} = 7$$ km which is how far Petra walked in two days. So he walked twice as far in four days, which is $$7 \times 2 = 14$$ km.

So the total distance Petra walked that week is $$6 + 14 = 20$$ km.

Alternatively, we can do the calculation in metres.
Since we know that $$1$$ km is equal to $$1000$$ m, then $$\frac{1}{2}$$ km is equal to $$500$$ m.
So $$3 \frac{1}{2}$$ km is equal to $$3 \times 1000 + 500 = 3500$$ m and $$2$$ km is equal to $$2 \times 1000=2000$$ m.

This means the total distance Petra walked is equal to $$2000 + 2000 + 2000 + 3500 + 3500 + 3500 + 3500 = 20000$$ m, which is $$20$$ km.