Order of operations, estimating and remainders
What this lesson covers
- Rounding, estimation, and using benchmarks support comparing numbers and checking whether findings are reasonable.
- Division can result in a remainder expressed as a whole number, fraction, or decimal.
- In expressions that have more than one operation, the order of operations is important.
- A mnemonic, such as GEMA: grouped, exponents, multiplicative (× and ÷), and additive (+ and −) can be used to remember the order of operations.
Two people, one sum, two answers
- Work out 3 + 4 × 2.
- Straight along the line you get 14.
- Somebody else gets 11. One of you is wrong.
| Method | Working | Gets |
|---|---|---|
| Left to right | 3 + 4 = 7, then 7 × 2 | 14 |
| By the rules | 4 × 2 = 8, then 3 + 8 | 11 |
The order everything follows
- Grouped, exponents, multiplicative, additive. GEMA.
- You may have met BEDMAS. It is the same order.
- GEMA is safer, because M and A each name a pair.
| Stands for | Meaning | |
|---|---|---|
| G | Grouped | anything inside brackets |
| E | Exponents | squares, cubes, roots |
| M | Multiplicative | × and ÷, left to right |
| A | Additive | + and −, left to right |
G: brackets go first, always
- Anything inside brackets is settled before anything else.
- 3 + 4 × 2 is 11. But (3 + 4) × 2 is 14.
- Brackets are how you overrule the order on purpose.
| Expression | Why | Answer |
|---|---|---|
| 3 + 4 × 2 | no brackets | 11 |
| (3 + 4) × 2 | brackets first | 14 |
| Same digits | different instruction |
E: then squares, cubes and roots
- An exponent is settled before any × or ÷ touches it.
- In 3² × 2, the 3² becomes 9 first.
- Then 9 × 2. Never 3 × 2 squared.
| Expression | Step | Becomes |
|---|---|---|
| 3² × 2 | square the 3 | 9 × 2 |
| then multiply | 18 | |
| Not | (3 × 2)² | 36 |
M: and here is where people slip
- × and ÷ do not take turns. They rank equally.
- So you go left to right. 20 ÷ 4 × 5 starts with 20 ÷ 4.
- That is 5, then 5 × 5 is 25. Not 1.
| Step | Why | Gives |
|---|---|---|
| 20 ÷ 4 × 5 | left to right | |
| 20 ÷ 4 | first, because it is leftmost | 5 |
| 5 × 5 | then this | 25 |
| Common error | doing 4 × 5 first gives | 1 |
A: exactly the same trap
- + and − also rank equally, so also left to right.
- 10 − 3 + 2 starts with 10 − 3, which is 7.
- Then 7 + 2 is 9. Adding first would give 5.
| Step | Why | Gives |
|---|---|---|
| 10 − 3 + 2 | left to right | |
| 10 − 3 | first | 7 |
| 7 + 2 | then | 9 |
| Common error | doing 3 + 2 first gives | 5 |
All four letters at once
- 2 + 3² × 2. No brackets, so start at E.
- 3² is 9, giving 2 + 9 × 2.
- M next: 9 × 2 is 18. Then A: 2 + 18 = 20.
| Expression | Step | Becomes |
|---|---|---|
| 2 + 3² × 2 | E: 3² = 9 | 2 + 9 × 2 |
| M: 9 × 2 = 18 | 2 + 18 | |
| A: add | 20 |
When the answer goes below zero
- 5 − 2 × 3. The multiplication is first.
- 2 × 3 is 6, so this is 5 − 6.
- Which is −1. Order of operations does not avoid negatives.
Now: knowing the answer before you get it
- A calculator will happily return a wrong answer.
- It cannot tell you that you typed 147 instead of 14.7.
- So decide roughly what it must be, first.
| Catches | |
|---|---|
| A rough guess | nothing much |
| An upper and lower bound | a misplaced decimal point |
| Nothing at all | nothing |
Trapping 14.7 × 5 between two easy sums
- 14.7 is between 14 and 15.
- 14 × 5 is 70. 15 × 5 is 75.
- So the answer must be between 70 and 75. It is 73.5.
| Benchmark | Compared | Value |
|---|---|---|
| 14 × 5 | too small | 70 |
| 15 × 5 | too big | 75 |
| So 14.7 × 5 | must be between them | 70 to 75 |
| Actual | 73.5 |
Rounding to any power of ten
- 4728 to the nearest 10 is 4730.
- To the nearest 100 it is 4700.
- To the nearest 1000 it is 5000. The 7 rounds the 4 up.
| Number | To the | Becomes |
|---|---|---|
| 4728 | nearest 10 | 4730 |
| 4728 | nearest 100 | 4700 |
| 4728 | nearest 1000 | 5000 |
And to any decimal place
- Same method, going the other way down the ladder.
- 3.14159 to a tenth is 3.1, to a hundredth 3.14.
- To a thousandth, 3.142. Look at the digit just past.
| Number | To the | Becomes |
|---|---|---|
| 3.14159 | nearest whole | 3 |
| 3.14159 | nearest tenth | 3.1 |
| 3.14159 | nearest hundredth | 3.14 |
| 3.14159 | nearest thousandth | 3.142 |
The zero you have to write
- Round 2.798 to the nearest hundredth.
- The 8 rounds 9 up to 10, which carries. You get 2.80.
- Write the zero. 2.8 is the same value but shows no hundredths.
| Answer | Shows | Verdict |
|---|---|---|
| 2.80 | hundredths, as asked | right |
| 2.8 | tenths only | marked wrong |
Dividing when it does not go evenly
- 327 ÷ 15. How many 15s fit into 327?
- 15 × 20 is 300, and one more 15 makes 315.
- So 21 of them, with 12 left over.
| Form | Working | Answer |
|---|---|---|
| Whole number | 15 × 21 = 315, 12 left over | 21 r 12 |
Three ways to write what is left over
- As a whole number: 21 remainder 12.
- As a fraction: 12 out of 15 is 4/5, so 21 4/5.
- As a decimal: 4/5 is 0.8, so 21.8. All three are right.
| Form | Working | Answer |
|---|---|---|
| Whole number | 15 × 21 = 315, 12 left over | 21 r 12 |
| Fraction | 12 out of 15 is 4/5 | 21 4/5 |
| Decimal | 4/5 is 0.8 | 21.8 |
Which one the question wants
- Buses for 327 people: you cannot run 0.8 of a bus. 22.
- Sharing $327: money wants the decimal.
- Cutting 327 cm: the fraction is exact. Read the context.
| Situation | Why | Use |
|---|---|---|
| Buses for 327 people | you cannot run 0.8 of a bus | 22 buses |
| Sharing $327 | money takes decimals | $21.80 |
| Cutting 327 cm | a fraction is exact | 21 4/5 cm |
Your turn: four questions
- Pause here and try all four.
- Question 2 is the left-to-right one. Take your time.
- Question 3 wants two decimal places in the answer.
| Question | |
|---|---|
| 1 | Work out 6 + 2 × 5. |
| 2 | Work out 18 ÷ 3 × 2. |
| 3 | Round 2.798 to the nearest hundredth. |
| 4 | Work out 50 ÷ 8, three ways. |
Show answer
1. 16. The multiplication goes first: 2 × 5 = 10, then 6 + 10. 2. 12. Left to right, so 18 ÷ 3 = 6 first, then 6 × 2. Doing 3 × 2 first gives 3, which is the error this lesson is built around. 3. 2.80. The 8 in the thousandths rounds the hundredths up from 9 to 10, which carries. Write the trailing zero: 2.8 is the same value but does not show hundredths. 4. 6 remainder 2, or 6 1/4, or 6.25.
The answers
- 6 + 2 × 5 is 16, because 2 × 5 goes first.
- 18 ÷ 3 × 2 is 12, left to right. Not 3.
- 2.798 to a hundredth is 2.80. And 50 ÷ 8 is 6.25.
| Answer | |
|---|---|
| 1 | 16 |
| 2 | 12 |
| 3 | 2.80 |
| 4 | 6 r 2, or 6 1/4, or 6.25 |
What to take away
- GEMA, and the M and A each cover a pair that runs left to right.
- Bracket an answer above and below before you calculate it.
- A remainder has three forms. The context picks one.
| Remember | |
|---|---|
| GEMA | grouped, exponents, multiplicative, additive |
| M and A | each name a pair, and pairs run left to right |
| Brackets | the only way to overrule the order |
| Estimate | bracket it above and below, then calculate |
| A remainder | whole, fraction or decimal, as the question needs |
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