Adding in Any Order

Pick the order that makes a ten first.

The same hops, the other way round

Put 2 marbles in a bag, then 9 more, and the bag holds 11. Put the 9 in first and then the 2, and it still holds 11. The same marbles went in; only the order changed.

On a number line, start at 0, hop 2 and then hop 9, and you land on 11. Hop 9 first and then 2, and you land on 11 again. So 2 + 9 = 9 + 2. Two numbers added in either order give the same total, and this rule is called the commutative property of addition.

015+2+911

2 + 9 is a hop of 2, then a hop of 9, and it lands on 11.

015+9+211

9 + 2 is the same two hops in the other order, and it lands on 11 too.

012345678910111129

2 + 9 = 11: the two rods laid end to end reach 11

Turn the train end to end

A rod of 2 and a rod of 9, laid end to end, reach 11. Half a turn puts the 9 first, and the ends land on 0 and 11 again: the same two rods make the same length.

Choose the easy order

Because the order never changes the total, you may add in whichever order is easiest. Counting on is quicker from the bigger number: 2 + 9 is nine counts, but 9 + 2 is only two.

With three numbers, look for two that make ten. In 7 + 38 + 3, the 7 and the 3 make 10, so add them first. Then 38 + 10 = 48, which is easier than 7 + 38 = 45 followed by 45 + 3.

Only adding works in any order. Taking away does not: 9 − 2 = 7, but 2 − 9 is not 7.

050+7+3+3848

The 7 and the 3 reach 10 first, and then the 38 goes on in one hop, landing on 48.

Worked example: Three Prices Added in the Easiest Order

Question Mei buys a book for $37, a bag for $48 and a lamp for $63. (a) How much do the book and the lamp cost together? (b) How much do the three things cost altogether?

  1. 1.Look at the ones digits of the three prices. 7 + 3 = 10, so 37 and 63 are a good pair to add first.

    As written$37 book$48 bag$63 lamp
    As written$37$48$63
    7 + 3 = 10, so 37 and 63 are a good pair.

  2. 2.(a) 30 + 60 = 90 and 7 + 3 = 10, so 37 + 63 = 90 + 10 = 100. The book and the lamp cost $100 together.

    As written$37 book$48 bag$63 lampPair first$37$6337 + 63 = 100
    As written$37$48$63Pair first$37$6337 + 63 = 100
    (a) 37 + 63 = 100. The book and the lamp cost $100 together.

  3. 3.Numbers can be added in any order and the total stays the same. So add the price of the bag last.

    As written$37 book$48 bag$63 lampPair first$37$63$4837 + 63 = 100
    As written$37$48$63Pair first$37$63$4837 + 63 = 100
    The order does not change the total, so the bag is added last.

  4. 4.(b) 100 + 48 = 148, so the three things cost $148 altogether. Check: 37 + 48 = 85 and 85 + 63 = 148.

    As written$37 book$48 bag$63 lampPair first$37$63$48$14837 + 63 = 100100 + 48 = 148
    As written$37$48$63Pair first$37$63$48$14837 + 63 = 100100 + 48 = 148
    (b) 100 + 48 = 148. The three things cost $148 altogether.

Answer: (a) $100; (b) $148

Common mistakes

  • Adding 37 and 63 to make 100 and then stopping. The total needs all three prices, so the $48 for the bag must still be added.
  • Thinking that 48 and 63 make 100 because 4 + 6 = 10. The ones digits 8 and 3 make another ten, so 48 + 63 is more than 100.

More adding and subtracting in your head problems, worked step by step →

Worked example: Nine Dartboard Numbers Added in Tens, and the Two a Player Missed

Question At a school fair, a dartboard has nine rings numbered 3, 7, 4, 8, 6, 2, 9, 1 and 5. Wei Ming throws 7 darts and hits 7 different rings. (a) Find the total of all nine numbers on the board. Pair numbers that make 10 to add them in your head. (b) The 7 numbers he hit add up to 34. What do the two numbers he missed add up to? (c) List every pair of numbers he could have missed.

  1. 1.Look for pairs that make 10: 3 + 7, 4 + 6, 8 + 2 and 9 + 1. That uses eight of the nine numbers, and only 5 is left without a partner.

    3748629153 + 7104 + 6108 + 2109 + 11055
    3748629153 + 7104 + 6108 + 2109 + 11055
    3 + 7, 4 + 6, 8 + 2 and 9 + 1 each make 10. Only 5 has no partner.

  2. 2.(a) Four tens and the 5: 10 + 10 + 10 + 10 + 5 = 45. The nine numbers add up to 45.

    3748629153 + 7104 + 6108 + 2109 + 11055(a) 10 + 10 + 10 + 10 + 5 = 45
    3748629153 + 7104 + 6108 + 2109 + 11055(a) 10 + 10 + 10 + 10 + 5 = 45
    (a) 10 + 10 + 10 + 10 + 5 = 45.

  3. 3.The 7 numbers he hit and the 2 numbers he missed are all nine numbers together. So the numbers he missed make up what is left when 34 is taken from 45.

    3748629153 + 7104 + 6108 + 2109 + 11055(a) 10 + 10 + 10 + 10 + 5 = 45All nine7 numbers hit: 34?45 in all
    3748629153 + 7104 + 6108 + 2109 + 11055(a) 10 + 10 + 10 + 10 + 5 = 45All ninehit: 34?45 in all
    The numbers hit and the numbers missed together make all nine: 45.

  4. 4.(b) Count up from 34 to 45: 34 + 6 = 40 and 40 + 5 = 45, so the gap is 6 + 5 = 11. The two numbers he missed add up to 11. Check: 34 + 11 = 45.

    3748629153 + 7104 + 6108 + 2109 + 11055(a) 10 + 10 + 10 + 10 + 5 = 45All nine7 numbers hit: 341145 in all34 + 6 = 40 and 40 + 5 = 45, so the gap is 11
    3748629153 + 7104 + 6108 + 2109 + 11055(a) 10 + 10 + 10 + 10 + 5 = 45All ninehit: 341145 in all34 + 6 = 40 and 40 + 5 = 45, so the gap is 11
    (b) Count up from 34: 34 + 6 = 40 and 40 + 5 = 45. The two numbers missed make 11.

  5. 5.Find two different numbers on the board that make 11, starting from the smallest. 1 would need 10, which is not on the board. Then come 2 + 9, 3 + 8, 4 + 7 and 5 + 6. After that the pairs repeat the other way round, as 6 + 5.

    3748629153 + 7104 + 6108 + 2109 + 11055(a) 10 + 10 + 10 + 10 + 5 = 45All nine7 numbers hit: 341145 in all34 + 6 = 40 and 40 + 5 = 45, so the gap is 111 would need 10, which is not on the board2 + 9, 3 + 8, 4 + 7 and 5 + 6 make 11
    3748629153 + 7104 + 6108 + 2109 + 11055(a) 10 + 10 + 10 + 10 + 5 = 45All ninehit: 341145 in all34 + 6 = 40 and 40 + 5 = 45, so the gap is 111 would need 10, which is not on the board2 + 9, 3 + 8, 4 + 7 and 5 + 6 make 11
    Two different numbers on the board that make 11: 1 would need 10, then 2 + 9, 3 + 8, 4 + 7 and 5 + 6.

  6. 6.(c) He could have missed 2 and 9, 3 and 8, 4 and 7, or 5 and 6.

    3748629153 + 7104 + 6108 + 2109 + 11055(a) 10 + 10 + 10 + 10 + 5 = 45All nine7 numbers hit: 341145 in all34 + 6 = 40 and 40 + 5 = 45, so the gap is 111 would need 10, which is not on the board2 + 9, 3 + 8, 4 + 7 and 5 + 6 make 11(c) 4 pairs he could have missed
    3748629153 + 7104 + 6108 + 2109 + 11055(a) 10 + 10 + 10 + 10 + 5 = 45All ninehit: 341145 in all34 + 6 = 40 and 40 + 5 = 45, so the gap is 111 would need 10, which is not on the board2 + 9, 3 + 8, 4 + 7 and 5 + 6 make 11(c) 4 pairs he could have missed
    (c) He missed 2 and 9, 3 and 8, 4 and 7, or 5 and 6.

Answer: (a) 45; (b) 11; (c) 2 and 9, 3 and 8, 4 and 7, or 5 and 6

Common mistakes

  • Adding only the four pairs, 10 + 10 + 10 + 10 = 40, and leaving out the 5. Every number on the board is counted once, whether it has a partner or not, so the total is 45.
  • Listing 1 and 10, or 6 and 5, as more pairs. 10 is not on the board, and 6 and 5 is the same pair as 5 and 6, so there are only 4 pairs.

More adding and subtracting in your head problems, worked step by step →

Practice Adding in Any Order in the app