Fruit Basket Ratio

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 $o=3\times9-3=24.$