# Thirty Clerks

## Outline Mathematics Word Problems

Here's a problem to tackle:

Sam paused in the doorway. "Well!" he exclaimed. "What a change from last time I was here. You must have at least thirty clerks now."

"No, not that many yet." Mike grinned. "If we had twice as many girls or three times as many boys, then we'd have thirty. But wait another year or two."

How many clerks were there?

Solution

### Solution

Sam paused in the doorway. "Well!" he exclaimed. "What a change from last time I was here. You must have at least thirty clerks now."

"No, not that many yet." Mike grinned. "If we had twice as many girls or three times as many boys, then we'd have thirty. But wait another year or two."

How many clerks were there?

(In the text below, some words are omitted. These have been underlined. Move the cursor over the omissions. See what happens.)

Let the number of girl clerks be g and the number of boy clerks be b. Then the total number of clerks is b + g - not quite thirty. But, if the number of girls was twice their present number, there would be 30 clerks:

b + 2g = 30.

Additionally, if there were three times as many boys as now, again the total would be 30:

3b + g,b,g = 30.

There are many ways to solve a system of two linear equations. Let's multiply the first equation by 3:

3b + 6,3,4,5,6,8g = 90,30,60,90.

Subtract the second equation from the latter:

5,3,4,5,6,8g = 60,30,60,90.

So the number of girls is g = 12,10,12,14,16,18,20. From the first equation, b = 30 - 2g, or b = 30 - 24 = 6,3,4,5,6,8.

Answer: There are 12 girls and 6 boys, 18,10,12,14,16,18,20 clerks in all.

### References

1. J. A. H. Hunter, Entertaining Mathematical Teasers, Dover Publications, 1983

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