Order of operations · applications

Applications: Order of Operations

10 question types · Primary 5 · each worked step by step with a figure that follows the steps

PSLE · GCSE Higher

01

One Item and Several of Another in One Total

strategyMultiply Before Adding

A school bag costs $48 and a file costs $12. Mrs Tan buys 1 school bag and 6 files. (a) Write one expression for the total cost. (b) Find the total cost.

Bag$48Files$12$12$12$12$12$126 × $12
There is one bag at $48 and there are six files at $12 each.
The bag costs $48. The files cost 6 lots of $12, which is 6 × 12.
step 1 of 4

Write the cost of the bag and the cost of the files in one line. Only the files are bought six times, so only the 12 is multiplied by 6.

  1. The bag costs $48. The files cost 6 lots of $12, which is 6 × 12.
  2. (a) One expression for the total cost is 48 + 6 × 12.
  3. There are no brackets, so multiply before adding: 6 × 12 = 72.
  4. (b) Now add: 48 + 72 = 120. The total cost is $120. Check: six files cost 12 + 12 + 12 + 12 + 12 + 12 = 72, and 72 + 48 = 120.

answer(a) 48 + 6 × 12; (b) $120

techniqueMultiply and Divide Before Adding and Subtracting · Two-Step Word Problems

examsPSLE · GCSE Higher

Common pitfalls

  • Adding first because the addition is written first: 48 + 6 = 54, then 54 × 12 = 648. Multiplication is done before addition, wherever it is written in the line.
  • Writing (48 + 6) × 12. These brackets would mean 54 things that each cost $12, but the bag does not cost $12.
02

Change from a Note After Several Purchases

strategyBrackets Around Everything That Is Spent

Ravi buys 3 notebooks at $4 each and a pen at $7. He pays with a $50 note. (a) Write one expression for his change. (b) Find his change.

The $50 note$4$4$4$7change ?
The three notebooks and the pen are all paid for out of the $50 note.
The money he spends is 3 × 4 + 7. All of it is taken away from $50, so it goes inside brackets.
step 1 of 4

The change is the note take away everything he spends. Put the spending inside brackets so that all of it is subtracted.

  1. The money he spends is 3 × 4 + 7. All of it is taken away from $50, so it goes inside brackets.
  2. (a) One expression for his change is 50 − (3 × 4 + 7).
  3. Work inside the brackets first, and multiply before adding: 3 × 4 = 12, then 12 + 7 = 19.
  4. (b) Subtract what he spends from the note: 50 − 19 = 31. His change is $31. Check: 19 + 31 = 50.

answer(a) 50 − (3 × 4 + 7); (b) $31

techniqueBrackets First · Multiply and Divide Before Adding and Subtracting · Two-Step Word Problems

examsPSLE · GCSE Higher

Common pitfalls

  • Leaving out the brackets: 50 − 3 × 4 + 7 = 50 − 12 + 7 = 45. This takes away the notebooks and then adds the price of the pen, so the change comes out too large.
  • Subtracting from left to right before multiplying: 50 − 3 = 47. The 3 is a number of notebooks, not a number of dollars, so it cannot be taken away from $50.
03

Two Collections Put Together and Shared Equally

strategyBrackets Before Dividing

Aini has 96 stickers and Ben has 48 stickers. They put all their stickers together and share them equally among 6 children. (a) How many stickers does each child get? (b) Ben writes 96 + 48 ÷ 6. What is the value of Ben's expression, and why is it not the answer?

Aini96Ben48All shared(96+48)÷6Ben writes96+48÷6
All the stickers are shared, so the adding comes first: (96 + 48) ÷ 6.
All the stickers are shared, so add them first. Brackets say that the adding comes first: (96 + 48) ÷ 6.
step 1 of 4

All the stickers are shared, so the two numbers must be added before the division. Brackets are the way to say that the addition comes first.

  1. All the stickers are shared, so add them first. Brackets say that the adding comes first: (96 + 48) ÷ 6.
  2. Work inside the brackets: 96 + 48 = 144.
  3. (a) Divide the total by 6: 144 ÷ 6 = 24. Each child gets 24 stickers. Check: 6 × 24 = 144.
  4. (b) Ben's expression has no brackets, so the division comes first: 48 ÷ 6 = 8, then 96 + 8 = 104. It shares only Ben's 48 stickers and leaves Aini's 96 stickers unshared.

answer(a) 24 stickers; (b) 104, because only Ben's 48 stickers are divided by 6

techniqueBrackets First · Multiply and Divide Before Adding and Subtracting

examsPSLE · GCSE Higher

Common pitfalls

  • Sharing only one collection and stopping, for example 96 ÷ 6 = 16. Each child gets a share of both collections, and 16 + 8 = 24.
  • Putting the brackets round the division, 96 + (48 ÷ 6). The division already comes first there, so these brackets change nothing and the value is still 104.
04

Weekly Savings, One Purchase and a Target

strategyA Bracket for the Money She Has

Mei saves $15 each week for 8 weeks. Then she spends $32 of her savings on a gift. She wants a bicycle that costs $180. (a) How much money does she have after buying the gift? (b) Write one expression for the amount she still needs, and find its value.

Bicyclehas ?needs ?$180Money she has8×15−32
She saves 8 × 15 dollars and then spends $32: 8 × 15 − 32.
Her savings are 8 × 15, and the gift takes $32 away. The money she has is 8 × 15 − 32.
step 1 of 4

First write the money she has as one expression. The amount she still needs is $180 take away all of that money, so the first expression goes inside brackets.

  1. Her savings are 8 × 15, and the gift takes $32 away. The money she has is 8 × 15 − 32.
  2. (a) Multiply first, then subtract: 8 × 15 = 120 and 120 − 32 = 88. She has $88.
  3. The amount she still needs is $180 take away all the money she has, so her money goes inside brackets: 180 − (8 × 15 − 32).
  4. (b) The brackets are worth 88, so 180 − 88 = 92. She still needs $92. Check: 88 + 92 = 180.

answer(a) $88; (b) 180 − (8 × 15 − 32) = $92

techniqueBrackets with All Four Operations · Multiply and Divide Before Adding and Subtracting

examsPSLE · GCSE Higher

Common pitfalls

  • Leaving out the brackets: 180 − 8 × 15 − 32 = 180 − 120 − 32 = 28. The gift made her money smaller, so she needs more because of it, not $32 less.
  • Giving $88 as the answer to part (b). $88 is the money she has. The money she still needs is 180 − 88.
05

The Perimeter of a Rectangle Written Two Ways

strategyTwice the Sum Equals the Sum of the Doubles

A rectangular garden is 28 m long and 17 m wide. A fence goes all the way round it. (a) Find the length of the fence with the expression 2 × (28 + 17). (b) Find it again with the expression 2 × 28 + 2 × 17. What do you notice?

28 m17 m2 × (length + width)2×(28+17)2×452 lengths + 2 widths2×28+2×17
One length and one width are half of the way round: 28 + 17 = 45 m.
One length and one width make half of the way round. The brackets come first: 28 + 17 = 45.
step 1 of 4

The way round a rectangle is two lengths and two widths. It can be written as twice one length and one width, or as two lengths added to two widths.

  1. One length and one width make half of the way round. The brackets come first: 28 + 17 = 45.
  2. (a) The whole way round is twice that: 2 × 45 = 90. The fence is 90 m long.
  3. The second expression has no brackets, so do both multiplications before the addition: 2 × 28 = 56 and 2 × 17 = 34.
  4. (b) Add: 56 + 34 = 90. The fence is 90 m long again. Both expressions count two lengths and two widths, so they are equal.

answer(a) 90 m; (b) 90 m, so the two expressions are equal

techniqueBrackets First · Multiply and Divide Before Adding and Subtracting · Area and Perimeter

examsPSLE · GCSE Higher

Common pitfalls

  • Working 2 × (28 + 17) as 2 × 28 + 17 = 73. The brackets say that the 2 multiplies the whole sum, so the width is doubled as well.
  • In 2 × 28 + 2 × 17, adding in the middle first: 28 + 2 = 30. Both multiplications are done before the addition, so the 28 and the 2 next to it are never added.
06

One Expression, Two Pupils, Two Answers

strategyDivide First, Then Work from Left to Right

Ali and Bea both work out 36 − 12 ÷ 4 + 2. Ali gets 8 and Bea gets 35. (a) What is the correct value? (b) Who is right, and how did the other pupil get a different answer?

Bea: division first36−12÷4+236−3+2Ali: left to right36−12÷4+2
There are no brackets, so the division comes first: 12 ÷ 4 = 3.
There are no brackets, so the division comes first: 12 ÷ 4 = 3. The expression becomes 36 − 3 + 2.
step 1 of 4

With no brackets, division is done before subtraction and addition. After that, subtraction and addition are done in the order they appear, from left to right.

  1. There are no brackets, so the division comes first: 12 ÷ 4 = 3. The expression becomes 36 − 3 + 2.
  2. Only subtraction and addition are left, so work from left to right: 36 − 3 = 33.
  3. (a) Then 33 + 2 = 35. The correct value is 35.
  4. (b) Bea is right. Ali worked from left to right from the start: 36 − 12 = 24, then 24 ÷ 4 = 6, then 6 + 2 = 8. He subtracted before he divided.

answer(a) 35; (b) Bea is right. Ali worked from left to right and got 8

techniqueMultiply and Divide Before Adding and Subtracting

examsPSLE · GCSE Higher

Common pitfalls

  • Adding before subtracting in 36 − 3 + 2: 3 + 2 = 5, then 36 − 5 = 31. Subtraction and addition are done in the order they appear, so the 3 is subtracted and then the 2 is added.
  • Working the whole expression from left to right, as Ali did. Division is done before subtraction even when the subtraction is written first.
07

One Pair of Brackets to Make a Statement True

strategyTry Each Place for the Brackets

The statement 4 + 6 × 7 − 2 = 34 is not true. One pair of brackets makes it true. (a) What is the value of 4 + 6 × 7 − 2 with no brackets? (b) Where do the brackets go?

The value must be 34.No brackets4+6×7−24+42−244
(a) With no brackets, multiply first: 6 × 7 = 42, then 4 + 42 − 2 = 44. That is not 34.
(a) With no brackets, multiply first: 6 × 7 = 42. Then 4 + 42 − 2 = 44, which is not 34.
step 1 of 4

Find the value with no brackets first. Then put the brackets round one addition or subtraction at a time, because brackets round the multiplication would change nothing.

  1. (a) With no brackets, multiply first: 6 × 7 = 42. Then 4 + 42 − 2 = 44, which is not 34.
  2. Try the brackets round 4 + 6: (4 + 6) × 7 − 2 = 10 × 7 − 2 = 68. That is too large.
  3. Try the brackets round 7 − 2: 4 + 6 × (7 − 2) = 4 + 6 × 5.
  4. (b) Multiply, then add: 6 × 5 = 30 and 4 + 30 = 34. The brackets go round 7 − 2, and the true statement is 4 + 6 × (7 − 2) = 34.

answer(a) 44; (b) 4 + 6 × (7 − 2) = 34

techniqueBrackets First · Multiply and Divide Before Adding and Subtracting

examsPSLE · GCSE Higher

Common pitfalls

  • Putting the brackets round 6 × 7. The multiplication is already done first, so the value stays 44.
  • After writing 4 + 6 × 5, adding first to get 10 × 5 = 50. The brackets are finished, but multiplication still comes before addition.
08

A Calculator That Works Strictly from Left to Right

strategyDivide and Multiply from Left to Right, Then Add

Jun's calculator does every operation in the order it is keyed in, from left to right. He keys in 18 + 42 ÷ 6 × 2. (a) What does his calculator show? (b) What is the correct value of 18 + 42 ÷ 6 × 2?

The calculator18+42÷6×260÷6×2The correct order18+42÷6×2
The calculator adds first, because the addition is keyed in first: 18 + 42 = 60.
The calculator adds first, because the addition is keyed in first: 18 + 42 = 60.
step 1 of 5

Follow the calculator first, one key at a time. Then use the correct order: division and multiplication from left to right, and addition last.

  1. The calculator adds first, because the addition is keyed in first: 18 + 42 = 60.
  2. (a) It carries on from left to right: 60 ÷ 6 = 10, then 10 × 2 = 20. The calculator shows 20.
  3. In the correct order, division and multiplication come before addition. Work them from left to right: 42 ÷ 6 = 7.
  4. Then 7 × 2 = 14. The expression is now 18 + 14.
  5. (b) Add last: 18 + 14 = 32. The correct value is 32.

answer(a) 20; (b) 32

techniqueMultiply and Divide Before Adding and Subtracting

examsPSLE · GCSE Higher

Common pitfalls

  • Multiplying before dividing, 6 × 2 = 12, and then trying 42 ÷ 12. Multiplication and division are done in the order they appear, from left to right, and the division appears first.
  • Accepting the 20 on the calculator. It added 18 + 42 first, but the 42 must be divided by 6 and multiplied by 2 before anything is added to 18.
09

Large and Small Boxes, Some Removed, the Rest Repacked

strategyOne Expression with All Four Operations

A fruit seller has 5 large boxes with 24 apples in each and 8 small boxes with 12 apples in each. He removes 36 bad apples and packs the rest into bags of 6. (a) How many good apples are there? (b) Write one expression for the number of bags, and find its value.

Large boxes24242424245 × 24 = 120Small boxes12121212121212128 × 12 = 96
The large boxes hold 5 × 24 = 120 apples and the small boxes hold 8 × 12 = 96.
The large boxes hold 5 × 24 = 120 apples and the small boxes hold 8 × 12 = 96 apples.
step 1 of 4

Count the apples in each kind of box, take away the bad ones, and then divide. All the good apples are divided by 6, so their expression goes inside brackets.

  1. The large boxes hold 5 × 24 = 120 apples and the small boxes hold 8 × 12 = 96 apples.
  2. (a) Add the two kinds of boxes and take away the bad apples: 120 + 96 − 36 = 180. There are 180 good apples.
  3. All the good apples are packed, so their expression goes inside brackets before the division: (5 × 24 + 8 × 12 − 36) ÷ 6.
  4. (b) The brackets are worth 180, so 180 ÷ 6 = 30. He packs 30 bags. Check: 30 × 6 = 180.

answer(a) 180 apples; (b) (5 × 24 + 8 × 12 − 36) ÷ 6 = 30 bags

techniqueBrackets with All Four Operations · Multiply and Divide Before Adding and Subtracting

examsPSLE · GCSE Higher

Common pitfalls

  • Leaving out the brackets: 5 × 24 + 8 × 12 − 36 ÷ 6 = 120 + 96 − 6 = 210. Without brackets only the 36 bad apples are divided by 6.
  • Adding the numbers of boxes first, 5 + 8 = 13, and multiplying by one box size. The large and small boxes hold different numbers of apples, so each kind is multiplied on its own.
10

A Think-of-a-Number Chain

strategyWrite the Chain with Brackets, Then Undo It in Reverse

Siti thinks of the number 9. She adds 6, multiplies the result by 4 and then subtracts 12. (a) Write her steps as one expression and find its value. (b) Tom does the same steps with a different number and ends with 44. What number did Tom think of?

Siti starts with 9(9+6)×4−12
The adding is done first, so it needs brackets: (9 + 6) × 4 − 12.
The adding is done first, so it needs brackets. The expression is (9 + 6) × 4 − 12.
step 1 of 5

The addition is done first, so it needs brackets. To find a starting number from the final number, undo the steps one at a time, starting from the last step.

  1. The adding is done first, so it needs brackets. The expression is (9 + 6) × 4 − 12.
  2. (a) Brackets first: 9 + 6 = 15. Then multiply: 15 × 4 = 60. Then subtract: 60 − 12 = 48. The value is 48.
  3. For Tom, undo the steps in reverse order. His last step was to subtract 12, so add 12: 44 + 12 = 56.
  4. Before that he multiplied by 4, so divide by 4: 56 ÷ 4 = 14.
  5. (b) His first step was to add 6, so subtract 6: 14 − 6 = 8. Tom thought of 8. Check: (8 + 6) × 4 − 12 = 56 − 12 = 44.

answer(a) (9 + 6) × 4 − 12 = 48; (b) 8

techniqueBrackets with All Four Operations · Brackets First

examsPSLE · GCSE Higher

Common pitfalls

  • Writing 9 + 6 × 4 − 12 without brackets. Its value is 9 + 24 − 12 = 21, because only the 6 is multiplied by 4.
  • Undoing Tom's steps in the order they were done, starting with subtracting 6. The last step done is the first step to undo, so begin by adding 12.
Mr. Chalk Drill the order of operations