Two purchases of the same two items

Two purchases of the same two items, with two different totals, hide the price of each item. Your child makes the count of one item match in both purchases, so the difference in the totals is the cost of the extra items of the other kind. The calculations are with money and masses in decimals, and the reasoning compares the two purchases one item at a time.

• Line up the two purchases, one above the other, with the same item in the same column.
• If one item's count already matches, the difference in totals belongs only to the other item.
• If no count matches, multiply a whole purchase, every count and its total, until one count does.
• Divide the difference in totals by the difference in counts to find one item's price.
• Put that price back into a purchase to find the other price, then answer what the question asks for.

In Question 1, three times the first purchase is 6 pens and 3 files for $29.40, so the 2 extra files cost $34.20 minus $29.40, which is $4.80, and one file is $2.40. Then 2 pens cost $9.80 minus $2.40, so a pen is $3.70. Both totals must come back: 6 times $3.70 plus 5 times $2.40 is $34.20. If only one total comes back, a price needs another look.

Question 1 is free: 2 pens and 1 file cost $9.80, 6 pens and 5 files cost $34.20, and no count matches yet, so your child takes the first purchase 3 times to match the pens. All five questions are Intermediate; the step where one count already matches is the one the lesson film on this page works through, so practice starts a step further on. The other four compare two people buying the same number of books, take one kit apart to compare it with items bought separately, find three box masses from the mass of each pair, and finish with the mass of a new bundle of tins and jars. Ask your child to say which item is matched, and why, before subtracting any totals.

Does my child need algebra for these questions? No. Matching one item's count and subtracting the totals is done with multiplication, subtraction and division, with each step explained in words.

Why multiply the whole purchase and not just one item? Buying a purchase twice doubles every item in it and its total. Doubling only one item would describe a purchase whose total the question never gives.

How do the three box masses work? Adding the three pair masses counts each box twice, so half of that sum is the mass of all three boxes. Taking away one pair then leaves the third box.