Year 6: Combinations of two variables
Part of the Year 6 algebra chapter. Chapter overview · whole chapter on one page
Combinations of two variables
Sometimes two letters stand for amounts of two things, and the amounts must add up to a total. For example, if a is the number of 2p coins and b is the number of 5p coins, then the total value is 2a + 5b pence.
To find every combination, we list every possible number of one item, in order, and work out the number of the other item each time. Only the lists where the number of each item is a whole number that fits the total count as combinations.
Make a table, so that no combination is missed and none is counted twice.
Key ideas
- A combination uses a whole number of each item.
- Start with 0 (or 1, if each item must be used at least once) and count up in order.
- For each choice, work out how much is left, and check that it can be made with the other item.
- Count the rows of the table to find the number of combinations.
Key words. A combination is one way of choosing an amount of each thing. To enumerate means to list every possibility in order, so that none is missed.
Method. List every choice in order, check each one, then count the ways.
- Write the total and the value of each item. One 2p coin is worth 2p. One 5p coin is worth 5p.12p from 2p and 5p items
- Choose how many 5p coins, starting at 0, and work out how much is left. The rest of the total must be made from 2p coins.the rest must be a whole number of the other item
- Keep the list in order, so that no way is missed or repeated. Trying each choice in turn gives a complete list.
Number of 2p coins Number of 5p coins Check 1 2 1 × 2 + 2 × 5 = 12 6 0 6 × 2 + 0 × 5 = 12 - Count the ways. Each row is one way.2 ways
- Write the answer. 2
Check. Each row adds to 12p, and no row is repeated. ✓
Watch out. Check that every row uses the right number of items. Do not forget the rows where one item is not used, if the question allows it.
Key words. A combination is one way of choosing an amount of each thing. To enumerate means to list every possibility in order, so that none is missed.
Method. List every choice in order, check each one, then count the ways.
- Write the total and the value of each item. One pen is worth 3p. One pencil is worth 5p.26p from 3p and 5p items
- Choose how many pencils, starting at 1, and work out how much is left. The rest of the total must be made from pens.the rest must be a whole number of the other item
- Keep the list in order, so that no way is missed or repeated. Trying each choice in turn gives a complete list.
Number of pens Number of pencils Check 2 4 2 × 3 + 4 × 5 = 26 7 1 7 × 3 + 1 × 5 = 26 - Count the ways. Each row is one way.2 ways
- Write the answer. 2
Check. Each row adds to 26p, and no row is repeated. ✓
Watch out. Check that every row uses the right number of items. Do not forget the rows where one item is not used, if the question allows it.
Key words. A combination is one way of choosing an amount of each thing. To enumerate means to list every possibility in order, so that none is missed.
Method. List every choice in order, check each one, then count the ways.
- Write the total and the value of each item. One red card is worth 9 points. One blue card is worth 4 points.49 points from 9 points and 4 points items
- Choose how many blue cards, starting at 1, and work out how much is left. The rest of the total must be made from red cards.the rest must be a whole number of the other item
- Keep the list in order, so that no way is missed or repeated. Trying each choice in turn gives a complete list.
Number of red cards Number of blue cards Check 1 10 1 × 9 + 10 × 4 = 49 5 1 5 × 9 + 1 × 4 = 49 - Count the ways. Each row is one way.2 ways
- Write the answer. 2
Check. Each row adds to 49 points, and no row is repeated. ✓
Watch out. Check that every row uses the right number of items. Do not forget the rows where one item is not used, if the question allows it.
Watch out
Check whether the question allows zero of one item. If it says “at least one of each”, start your list at 1, not 0.
Exam tip
Make a table with the two columns and a check column. Stop when the first item is too big to fit the total.
Exercise 8F Combinations and systems
Non-calculatorKey words. A combination is one way of choosing an amount of each thing. To enumerate means to list every possibility in order, so that none is missed.
Method. List every choice in order, check each one, then count the ways.
- Write the total and the value of each item. One 5p coin is worth 5p. One 10p coin is worth 10p.25p from 5p and 10p items
- Choose how many 10p coins, starting at 0, and work out how much is left. The rest of the total must be made from 5p coins.the rest must be a whole number of the other item
- Keep the list in order, so that no way is missed or repeated. Trying each choice in turn gives a complete list.
Number of 5p coins Number of 10p coins Check 1 2 1 × 5 + 2 × 10 = 25 3 1 3 × 5 + 1 × 10 = 25 5 0 5 × 5 + 0 × 10 = 25 - Count the ways. Each row is one way.3 ways
- Write the answer. 3
Check. Each row adds to 25p, and no row is repeated. ✓
Watch out. Check that every row uses the right number of items. Do not forget the rows where one item is not used, if the question allows it.
Key words. A combination is one way of choosing an amount of each thing. To enumerate means to list every possibility in order, so that none is missed.
Method. List every choice in order, check each one, then count the ways.
- Write the total and the value of each item. One 2p coin is worth 2p. One 10p coin is worth 10p.30p from 2p and 10p items
- Choose how many 10p coins, starting at 1, and work out how much is left. The rest of the total must be made from 2p coins.the rest must be a whole number of the other item
- Keep the list in order, so that no way is missed or repeated. Trying each choice in turn gives a complete list.
Number of 2p coins Number of 10p coins Check 5 2 5 × 2 + 2 × 10 = 30 10 1 10 × 2 + 1 × 10 = 30 - Count the ways. Each row is one way.2 ways
- Write the answer. 2
Check. Each row adds to 30p, and no row is repeated. ✓
Watch out. Check that every row uses the right number of items. Do not forget the rows where one item is not used, if the question allows it.
Key words. A combination is one way of choosing an amount of each thing. To enumerate means to list every possibility in order, so that none is missed.
Method. List every choice in order, check each one, then count the ways.
- Write the total and the value of each item. One adult is worth 7 points. One child is worth 3 points.40 points from 7 points and 3 points items
- Choose how many children, starting at 1, and work out how much is left. The rest of the total must be made from adults.the rest must be a whole number of the other item
- Keep the list in order, so that no way is missed or repeated. Trying each choice in turn gives a complete list.
Number of adults Number of children Check 1 11 1 × 7 + 11 × 3 = 40 4 4 4 × 7 + 4 × 3 = 40 - Count the ways. Each row is one way.2 ways
- Write the answer. 2
Check. Each row adds to 40 points, and no row is repeated. ✓
Watch out. Check that every row uses the right number of items. Do not forget the rows where one item is not used, if the question allows it.
Key words. A pair is two numbers that together make an equation true. Positive whole numbers are 1, 2, 3, 4 and so on (no zero, fractions or decimals).
Method. Work through the possible values of one letter, in order, and work out the other letter each time.
- Write the equation. We are looking for pairs that make it true.3x + 2y = 12
- Start with x = 1 and work out y. Keep only the values that give a whole number. Keep going, one value at a time, so that no pair is missed.
x y Check 2 3 3 × 2 + 2 × 3 = 12 - Stop when there are no more pairs with positive numbers. Once a number would be below 1, there is nothing left to find.the last pair is 2 and 3
- Count the pairs. Each row of the table is one pair.1 pairs
- Write the answer. 1
Check. Every row was checked in the equation. The table has 1 rows and no pair is repeated. ✓
Watch out. Start from 1, not from 0, for positive whole numbers. Make sure you have not counted a pair twice.
Key words. A combination is one way of choosing an amount of each thing. To enumerate means to list every possibility in order, so that none is missed.
Method. List every choice in order, check each one, then count the ways.
- Write the total and the value of each item. One apple is worth 25p. One banana is worth 15p.150p from 25p and 15p items
- Choose how many bananas, starting at 1, and work out how much is left. The rest of the total must be made from apples.the rest must be a whole number of the other item
- Keep the list in order, so that no way is missed or repeated. Trying each choice in turn gives a complete list.
Number of apples Number of bananas Check 3 5 3 × 25 + 5 × 15 = 150 - Count the ways. Each row is one way.1 ways
- Write the answer. 1
Check. Each row adds to 150p, and no row is repeated. ✓
Watch out. Check that every row uses the right number of items. Do not forget the rows where one item is not used, if the question allows it.
Key words. A combination is one way of choosing an amount of each thing. To enumerate means to list every possibility in order, so that none is missed.
Method. List every choice in order, check each one, then count the ways.
- Write the total and the value of each item. One 2p coin is worth 2p. One 10p coin is worth 10p.16p from 2p and 10p items
- Choose how many 10p coins, starting at 1, and work out how much is left. The rest of the total must be made from 2p coins.the rest must be a whole number of the other item
- Keep the list in order, so that no way is missed or repeated. Trying each choice in turn gives a complete list.
Number of 2p coins Number of 10p coins Check 3 1 3 × 2 + 1 × 10 = 16 - Count the ways. Each row is one way.1 ways
- Write the answer. 1
Check. Each row adds to 16p, and no row is repeated. ✓
Watch out. Check that every row uses the right number of items. Do not forget the rows where one item is not used, if the question allows it.
Key words. A combination is one way of choosing an amount of each thing. To enumerate means to list every possibility in order, so that none is missed.
Method. Start with all the counters as red, then look at what the blue ones add.
- Write the two clues. One clue is about the number of counters, the other about the points.r + b = 12 and 2r + 3b = 30
- If all 12 counters were red, the score would be 12 × 2 = 24. This is the starting score.24 points
- Each blue counter gives 1 more point than a red one. So the extra points come from the blue counters.3 - 2 = 1 extra point each
- Work out how many blue counters are needed. The score needs 6 more points.30 - 24 = 6, so b = 6 × 1 = 6
- Write the answer. 6
Check. 6 blue counters give 18 points and 6 red counters give 12 points, a total of 30. ✓
Watch out. Use the difference between the two counters, not the total of 12.
Key words. A combination is one way of choosing an amount of each thing. To enumerate means to list every possibility in order, so that none is missed.
Method. List every choice in order, check each one, then count the ways.
- Write the total and the value of each item. One 5p coin is worth 5p. One 10p coin is worth 10p.40p from 5p and 10p items
- Choose how many 10p coins, starting at 1, and work out how much is left. The rest of the total must be made from 5p coins.the rest must be a whole number of the other item
- Keep the list in order, so that no way is missed or repeated. Trying each choice in turn gives a complete list.
Number of 5p coins Number of 10p coins Check 2 3 2 × 5 + 3 × 10 = 40 4 2 4 × 5 + 2 × 10 = 40 6 1 6 × 5 + 1 × 10 = 40 - Count the ways. Each row is one way.3 ways
- Write the answer. 3
Check. Each row adds to 40p, and no row is repeated. ✓
Watch out. Check that every row uses the right number of items. Do not forget the rows where one item is not used, if the question allows it.
Key words. A combination is one way of choosing an amount of each thing. To enumerate means to list every possibility in order, so that none is missed.
Method. List the ways to make 12p with at least one of each coin, then count them.
- Try 1, 2 and 3 of the 3p coin, and see how much is left for the 2p coins. With at least one 2p coin, the 3p coins cannot use all the money.12 - 3 = 9, 12 - 6 = 6, 12 - 9 = 3
- Keep only the rows where the rest is a whole number of 2p coins. The rest must be even, so it can be split into 2p coins.9 is odd, 6 ÷ 2 = 3, and 3 is odd
- Count the ways. There is only one row that works.one way: 3 × 2p and 2 × 3p
- Write the answer. Sam is wrong, because there are not 4 ways.No
Check. 3 × 2 + 2 × 3 = 12, and no other row works. ✓
Watch out. Count only the rows that really work. Listing all the rows helps you check.
Key words. A combination is one way of choosing an amount of each thing. To enumerate means to list every possibility in order, so that none is missed.
Method. List every choice in order, check each one, then count the ways.
- Write the total and the value of each item. One pencil is worth 3p. One rubber is worth 7p.30p from 3p and 7p items
- Choose how many rubbers, starting at 1, and work out how much is left. The rest of the total must be made from pencils.the rest must be a whole number of the other item
- Keep the list in order, so that no way is missed or repeated. Trying each choice in turn gives a complete list.
Number of pencils Number of rubbers Check 3 3 3 × 3 + 3 × 7 = 30 - Count the ways. Each row is one way.1 ways
- Write the answer. 1
Check. Each row adds to 30p, and no row is repeated. ✓
Watch out. Check that every row uses the right number of items. Do not forget the rows where one item is not used, if the question allows it.
Key words. A combination is one way of choosing an amount of each thing. To enumerate means to list every possibility in order, so that none is missed.
Method. Compare the number of 2p coins with the number of 5p coins.
- Write the two clues. One clue is the number of coins, the other is the value.a + b = 20 and 2a + 5b = 70
- If all 20 coins were 2p, the total would be 40p. This is the starting total.20 × 2 = 40
- Each 5p coin adds 3p more than a 2p coin. So the extra money comes from the 5p coins.5 - 2 = 3
- Find how many 5p coins give the extra 30p. The total needs 70p, so 70 - 40 = 30 more.30 ÷ 3 = 10
- Write the answer. 10
Check. 10 coins of 5p = 50p and 10 coins of 2p = 20p, a total of 70p, from 20 coins altogether. ✓
Watch out. Every 5p coin replaces a 2p coin, so it adds only 3p more.
Key words. A combination is one way of choosing an amount of each thing. To enumerate means to list every possibility in order, so that none is missed.
Method. Compare with 10 coins of 20p, then find how many 50p coins add the extra money.
- Write the two clues. One clue is the number of coins, the other is the value.a + b = 10 and 50a + 20b = 260
- If all 10 coins were 20p, the total would be 200p. This is the starting total.10 × 20 = 200
- Each 50p coin adds 30p more than a 20p coin. So the extra money comes from the 50p coins.50 - 20 = 30
- Find how many 50p coins give the extra 60p. The total needs 260p, so 260 - 200 = 60 more.60 ÷ 30 = 2
- Write the answer. 2
Check. 2 coins of 50p and 8 coins of 20p give 100p + 160p = 260p, from 10 coins. ✓
Watch out. Check that both clues are true: the number of coins and the total value.