Two bags contain some numbers, bag A and bag B. Three of the numbers are moved from bag A into bag B. The sum of the numbers in bag B is now four times the sum of the numbers in bag A. Work out which three numbers were moved.
Before we can find out which numbers were moved, we need to calculate the original sum in both bag A and bag B. Bag A has numbers three, four, six, six, and eight. I know six plus six is 12, and 12 plus eight is 20. Three plus four is seven, and 20 plus seven equals 27. Bag A started with a sum of 27. And now bag B, 10 plus six plus two plus three plus seven. Three plus seven is 10. 10 plus 10 is 20, and six plus two is eight. 10 plus 10 is 20 plus eight is 28. Bag B started with a sum of 28.
We have subtracted three numbers with an unknown sum from bag A, and we’ve added that same amount to bag B. We want to let the sum of the three numbers moved, be equal to 𝑥. We would then be able to say 27 minus 𝑥 and 28 plus 𝑥. We also know that the sum of the numbers in bag B is now four times the amount in bag A. The sum in bag B is equal to four times the sum in bag A. We can substitute 28 plus 𝑥 in for the sum of all the numbers in bag B. And we can substitute 27 minus 𝑥 in for the sum of all the values in bag A.
We’ll give ourselves a little bit more room. And our statement says 28 plus 𝑥 equals four times 27 minus 𝑥. We need to expand our multiplication across the brackets, four times 27, and then four times negative 𝑥. Four times 27 equals 108, and four times negative 𝑥 equals negative four 𝑥. Bring down what was on the left-hand side. In order to solve for 𝑥, we need to get 𝑥 on the same side of the equation.
And we can do that by adding four 𝑥 to both sides. On the right-hand side, we have negative four 𝑥 plus four 𝑥, which cancels out, leaving us with 108. On the left-hand side, we add our four 𝑥 and our 𝑥 to get five 𝑥. And we bring down the 28. Now, we subtract 28 from both sides. 108 minus 28 equals 80. And then, we divide both sides by five. 80 divided by five equals 16. 𝑥 equals 16.
Remember that 𝑥 equals the sum of the three numbers we’ve moved, and that means we need to look in bag A and define three numbers whose sum is 16. If we add eight and six together, that’s 14. But we don’t have a two, so that’s too much. If we add six and six together we have 12. 12 plus four equals 16. There’s no other combination of three numbers in bag A which would sum to 16. So, we say that the numbers six, six and four were moved from bag A to bag B. These three values sum to 16.
There’s one final thing we want to check. We want to make sure that if we multiplied the sum of the values in bag A by four, we would get the sum of the values in bag B. 27 minus 16 is 11. 8 plus three is 11. So, the new sum for bag A is 11. 28 plus 16 is 44. The new sum for bag B is 44. And 11 times four does equal 44, which means we’ve correctly worked out that the three numbers that were moved were six, six, and four.