Well Read
If you know how many books each boy, girl and teddy bear borrowed from the library, can you work out the number of girls?
Age
14 to 16
Challenge level
Problem
In Miss Quaffley's class, one third of the pupils take a teddy bear to school.
Last term, each boy borrowed 12 books from the library, each girl borrowed 17 books, and each teddy bear borrowed 9 books.
In total 305 books were borrowed.
How many girls are there in the class?
If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
Student Solutions
Answer: 7 girls
$b$ boys and $g$ girls, so $\frac{b+g}{3}$ teddy bears.
Books taken out: $$\begin{align}12b+17g+9\times\tfrac{b+g}{3}&=305\\
\Rightarrow 12b+17g+3(b+g)&=305\\ \Rightarrow15b+20g&=305\end{align}$$
Divide through by $5$: $$3b+4g=61$$ $b$ and $g$ must be whole numbers
| $3b$ | $61-3b$ | Divisible by $4$? |
| $3$ | $58$ | no |
| $6$ | $55$ | no |
| even multiples of $3$ give odd numbers so only try odd multiples of $3$ |
||
| $9$ | $52$ | yes, $4\times13$; $3$ boys and $13$ girls |
| $15$ | $46$ | no |
| $21$ | $40$ | yes, $4\times10$; $7$ boys and $10$ girls |
| $27$ | $34$ | no |
| go up/down in $12$s (multiple of $4$) not $6$s to hit the multiples of $4$ |
||
| $33$ | $28$ | yes, $4\times7$; $11$ boys and $7$ girls |
| $45$ | $16$ | yes, $4\times4$; $15$ boys and $4$ girls |
| $57$ | $4$ | yes, $4\times1$; $19$ boys and $1$ girl |
Also need the number of teddy bears to be an integer so the total number of students is a multiple of $3$
$3+13=16$ no
$7 + 10 = 17$ no
$11 +7=18$ yes
$15+4=19$ no
$19+1=20$ no
So the number of girls is $7$.