Question

The ratio of the number of oranges to the number of apples is 1 : 3.21 oranges were added and the ratio became 4 : 5. How many fruitswere there initially?

Answer 1

There were 15 oranges initially.

There were 45 apples initially.

Hence, there were (15 + 45) = 60 fruits there initially.

Step-by-step explanation

Let the number of oranges be x

Let the number of apples be y

The ratio of the number of oranges to the number of apples is 1 : 3 implies:

`\begin{gathered} x\colon y=1\colon3 \\ (x)/(y)=(1)/(3) \\ \text{Cross multiply} \\ y*1=x*3 \\ y=3x----i \end{gathered}`

If 21 oranges were added and the ratio became 4 : 5, this implies:

`\begin{gathered} (x+21)\colon y=4\colon5 \\ (x+21)/(y)=(4)/(5) \\ \text{Cross multiply} \\ 5(x+21)=4* y \\ 5x+105=4y----ii \end{gathered}`

To know how many fruits were there initially, solve the system of the equations:

`\begin{gathered} \text{Substitute }y=3x\text{ into }ii \\ 5x+105=4(3x) \\ 5x+105=12x \\ \text{Combine the like terms} \\ 12x-5x=105 \\ 7x=105 \\ \text{Divide both sides by 7} \\ (7x)/(7)=(105)/(7) \\ x=15 \end{gathered}`

x = 15 implies there were 15 oranges initially.

To get y, substitute x = 15 into equation (i):

`\begin{gathered} y=3x----i \\ y=3*15 \\ y=45 \end{gathered}`

y = 45 implies there were 45 apples initially.

Hence, there were (15 + 45) = 60 fruits there initially.