The contents of a certain box consist of $14$ apples and $23$ oranges. How many oranges must be removed from the box so that $70\%$ of the pieces of fruit in the box will be apples?

### Concept & Logic In mixture or ratio problems where one component is removed, the other component remains entirely constant in absolute quantity. Here, the number of apples does not change. The fixed $14$ apples must represent the new target percentage ($70\%$) of the *new total* fruit count. $$\text{Constant Item} = \text{Target Percentage} \times \text{New Total}$$ ### Step-by-step Solution **Given:** Initial Apples = $14$ Initial Oranges = $23$ Initial Total = $37$ **Step 1: Set up the percentage equation** We want apples to be $70\%$ of the *new* total amount of fruit. $0.70 \times \text{New Total} = 14$ **Step 2: Solve for the New Total** $\text{New Total} = \frac{14}{0.70}$ $\text{New Total} = \frac{140}{7} = 20$ **Step 3: Determine the new number of oranges** If the new total is $20$ fruits, and $14$ of them are apples, the rest must be oranges. New Oranges $= 20 - 14 = 6$ **Step 4: Calculate the removed oranges** Initial Oranges ($23$) - New Oranges ($6$) = $17$ oranges removed. ### Exam Strategy & Shortcut Use fractional equivalencies for lightning-fast solving. $70\%$ is equivalent to the fraction $\frac{7}{10}$. This means apples represent $7$ parts out of a total $10$ parts. Since we know we have $14$ apples, those $7$ parts equal $14$. Therefore, $1$ part = $2$. The total fruit is $10$ parts, so the new total is $10 \times 2 = 20$. Initial total was $37$. New total is $20$. Difference is $17$ oranges. Solved mentally in 10 seconds. ### Common Pitfall A frequent mistake is calculating $70\%$ of the *old* total ($37$) to find the new number of oranges, which mathematically fails because the total itself shrinks as you remove items. Always base the percentage on the unknown *new* total. ### Final Answer **Therefore, the correct answer is 17.**