There were 225 apples and 253 pears in Box A. There were 260 apples and 212 pears in Box B. Mr. Chu moved some apples and pears from Box B to Box A. In the end, 50% of the fruits i... There were 225 apples and 253 pears in Box A. There were 260 apples and 212 pears in Box B. Mr. Chu moved some apples and pears from Box B to Box A. In the end, 50% of the fruits in Box A and 70% of the fruits in Box B were apples. How many fruits did Mr. Chu move from Box B to Box A?
Understand the Problem
The question is a word problem about fruits (apples and pears) being moved between two boxes, Box A and Box B. We need to determine the total number of fruits that were moved from Box B to Box A based on the given initial quantities, final percentages of apples in each box, and the fact that Mr. Chu moved some fruits between the boxes.
Answer
422
Answer for screen readers
422
Steps to Solve
- Calculate the initial total fruits and apples in each box
Box A: Total fruits $= 225 \text{ apples} + 253 \text{ pears} = 478$ Number of apples $= 225$
Box B: Total fruits $= 260 \text{ apples} + 212 \text{ pears} = 472$ Number of apples $= 260$
- Define variables
Let $a$ be the number of apples moved from Box B to Box A. Let $p$ be the number of pears moved from Box B to Box A. Total fruits moved from Box B to Box A is $a + p$.
- Express the final number of apples and total fruits in each box after the transfer
Box A: Total fruits $= 478 + a + p$ Number of apples $= 225 + a$
Box B: Total fruits $= 472 - (a + p)$ Number of apples $= 260 - a$
- Use the given percentages to form equations
In Box A, 50% of the fruits are apples: $$ \frac{225 + a}{478 + a + p} = 0.5 $$ $$ 225 + a = 0.5(478 + a + p) $$ $$ 450 + 2a = 478 + a + p $$ $$ a - p = 28 \qquad (1) $$
In Box B, 70% of the fruits are apples: $$ \frac{260 - a}{472 - (a + p)} = 0.7 $$ $$ 260 - a = 0.7(472 - a - p) $$ $$ 260 - a = 330.4 - 0.7a - 0.7p $$ $$ -0.3a + 0.7p = 70.4 \qquad (2) $$ Multiply equation (2) by 10: $$ -3a + 7p = 704 \qquad (3) $$
- Solve the system of equations
From equation (1), $a = p + 28$. Substitute this into equation (3): $$ -3(p + 28) + 7p = 704 $$ $$ -3p - 84 + 7p = 704 $$ $$ 4p = 788 $$ $$ p = 197 $$ Now, find $a$: $$ a = p + 28 = 197 + 28 = 225 $$
- Calculate the total number of fruits moved
Total fruits moved $= a + p = 225 + 197 = 422$
422
More Information
The problem involves setting up a system of equations based on the percentages of apples in the two boxes after some fruits were moved. Solving this system gives the number of apples and pears that were moved, and their sum gives the final answer.
Tips
A common mistake is incorrectly setting up the equations based on the percentages. For example, confusing which box has 50% apples and which has 70%. Another mistake is making errors in the algebraic manipulation when solving the system of equations.
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