Fruit Basket Ratio
From two ratios of apples and oranges in a fruit bowl, can you work out how many fruits there are?
Problem
My fruit basket contains apples and oranges.
The ratio of apples to oranges in the basket is 3 : 8.
When I remove one apple the ratio changes to 1 : 3.
How many oranges are in the basket?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
This problem is taken from the UKMT Mathematical Challenges.
Student Solutions
Answer: 24
Using equivalent ratios
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Using multiples
At the beginning, there are 8 oranges for every 3 apples, so the number of oranges must be a multiple of 8.
At the end, there are 3 oranges for every apple, so the number of oranges must be a multiple of 3.
The number of oranges does not change, so it must be a multiple of 8 and a multiple of 3. Common multiples of 3 and 8 are 24, 48, ... so we could try these as the number of oranges.
3 : 8 is equivalent to 9 : 24
1 : 3 is equivalent to 8 : 24
9 $-$ 1 = 8, so there were 9 apples and 24 oranges to begin with.
Using algebra
Let $a$ be the number of apples and $o$ be the number of oranges at the beginning. Then we have that the ratio $a:o$ is $3:8$ and the ratio $a-1:o$ is $1:3.$
These give the simultaneous equations $8a=3o$ and $3(a-1)=o\Rightarrow 3a-3=o$
Solving by elimination
Multiplying both sides of $3a-3=o$ by $3$ gives $9a-9=3o$. Subtracting the equation $8a=3o$ from $9a-9=3o$ gives $$\begin{align}9a-9-8a&=3o-3o\\\Rightarrow a-9&=0\\\Rightarrow a&=9\end{align}$$
Substituting $a=9$ into $3a-3=o$ gives $o=3\times9-3=24.$
Solving by substitution
Substituting $o=3a-3$ into $8a=3o$ gives $$\begin{align}8a&=3(3a-3)\\
\Rightarrow8a&=9a-9\\
\Rightarrow0&=a-9\\
\Rightarrow9&=a\end{align}$$
Substituting $a=9$ into $3a-3=o$ gives $o=3\times9-3=24.$