CEMC Pascal, 2015

Algebra Difficulty 1.4 Junior Find the answer Canada

Jim wrote the sequence of symbols $\heartsuit\,\spadesuit\,\spadesuit\,\heartsuit\,\diamondsuit\,\heartsuit\,\heartsuit\,\heartsuit\,\diamondsuit$ a total of 50 times. How many more $\heartsuit$ symbols than $\spadesuit$ symbols did he write?

Pick one

Solution

Solution 1

The sequence of symbols includes 5 $\heartsuit$ ’s and 2 $\spadesuit$ ’s.

This means that, each time the sequence is written, there are $5-2=3$ more $\heartsuit$ ’s written than $\spadesuit$ ’s.

When the sequence is written 50 times, in total there are $50 \cdot 3 = 150$ more $\heartsuit$ ’s written than $\spadesuit$ ’s.

Solution 2

The sequence of symbols includes 5 $\heartsuit$ ’s and 2 $\spadesuit$ ’s.

When the sequence is written 50 times, there will be a total of $50 \cdot 5 = 250$ $\heartsuit$ ’s written and a total of $50 \cdot 2 = 100$ $\spadesuit$ ’s written.

This means that there are $250-100 = 150$ more $\heartsuit$ ’s written than $\spadesuit$ ’s.

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.