Math 7 · Divisibility and decimals
Tests that save a division, and decimals in their place
A divisibility rule tells you whether one number divides evenly into another without doing the division, and each rule works for a reason you can see in place value. The rest of the unit is decimals: calculating with them, and putting numbers in order.
- 1. The words, first
- 2. Divisibility rules, and why they work
- 3. Decimals: four operations, one order
- 4. Comparing and ordering
- 5. What costs marks
The words, first
The idea: Most of this page is about place value: what a digit is worth because of where it sits.
| Word | What it means |
|---|---|
| Divisible | One number is divisible by another if dividing leaves no remainder. 24 ÷ 6 = 4 exactly, so 24 is divisible by 6. |
| Multiple | A number times a whole number. 24 is a multiple of 6, and 999 is a multiple of 9. |
| Digit sum | The digits of a number added together. For 2364 it is 2 + 3 + 6 + 4 = 15. |
| Place value | What a digit is worth because of its position. In 3.475 the 4 is 4 tenths and the 5 is 5 thousandths. |
| Order of operations | Brackets first. Then multiply and divide, left to right. Then add and subtract, left to right. |
| Estimate | A rough answer from rounded numbers. It shows where the decimal point belongs. |
| Numerator, denominator | The top and bottom of a fraction. A fraction is a division: 3/8 means 3 ÷ 8. |
| Benchmark | An easy number to compare with, such as 0, ½ or 1. |
Divisibility rules, and why they work
The idea: Each rule looks at only part of the number. The rest is tens, hundreds or thousands that already divide evenly, so it cannot leave a remainder.
| Divisible by | Test | Why it works |
|---|---|---|
| 2 | The last digit is even | Every ten divides by 2 |
| 5 | The last digit is 0 or 5 | Every ten divides by 5 |
| 10 | The last digit is 0 | The number is a whole count of tens |
| 4 | The last two digits divide by 4 | Every hundred does: 100 = 4 × 25 |
| 8 | The last three digits divide by 8 | Every thousand does: 1000 = 8 × 125 |
| 3 | The digit sum divides by 3 | See below |
| 9 | The digit sum divides by 9 | See below |
| 6 | It passes the tests for 2 and 3 | 6 = 2 × 3 |
Why the digit sum works. 10 is 9 + 1, 100 is 99 + 1 and 1000 is 999 + 1. So 2364 = 2 × 999 + 3 × 99 + 6 × 9 + (2 + 3 + 6 + 4). Everything before the bracket is a multiple of 9, so only the digit sum can leave a remainder — when dividing by 9, and by 3, which goes into 9.
Worked example. Can 2364 stickers be shared equally among 2, 3, 4, 5, 6, 8, 9 or 10 people?
- 2, 5 and 10: the last digit, 4, is even but not 0 or 5. Yes for 2 only.
- 4 and 8: 64 divides by 4, so yes for 4. For 8, take three digits: 364 ÷ 8 = 45.5, so no.
- 3, 9 and 6: the digit sum, 15, divides by 3 but not by 9. Passing 2 and 3 means 6 works too.
So 2, 3, 4 or 6 people. Check one: 2364 ÷ 6 = 394 ✓. The trap: 64 on its own divides by 8, so testing 8 with two digits gets it wrong.
Why nothing can be divided by zero. 12 ÷ 3 = 4 because 3 × 4 = 12. So 12 ÷ 0 would need a number that makes 12 when multiplied by 0, and there is none: anything times 0 is 0. (0 ÷ 12 is fine: it is 0.)
Decimals: four operations, one order
The idea: Follow the order of operations, work out the digits as if the point were not there, then use an estimate to put the point back.
6.4 − 1.2 × 3 + 4.5 ÷ 0.9 = 6.4 − 3.6 + 5 = 7.8
Multiply and divide before adding and subtracting. Going straight from left to right gives about 22.3.
Dividing by a decimal. 4.5 ÷ 0.9 asks how many 0.9s fit into 4.5. Multiply both numbers by 10 and the answer does not change: 45 ÷ 9 = 5.
Placing the point. For 2.8 × 4.1, work out 28 × 41 = 1148. The estimate 3 × 4 = 12 says the answer is near 12, so it is 11.48.
Worked example. Three granola bars cost $1.35 each and a juice costs $2.49. How much change do you get from $10?
10 − (3 × 1.35 + 2.49) = 10 − (4.05 + 2.49) = 10 − 6.54 = 3.46
The change is $3.46. Estimate: a bit over $4 plus about $2.50 leaves about $3.50 ✓. Dividing by more than one digit, or multiplying by more than two, is calculator work in Grade 7 — but the estimate is still your job.
Comparing and ordering
The idea: Write every number the same way — usually as decimals with the same number of places — then compare digit by digit from the left.
Longer is not bigger. Is 0.405 or 0.45 greater? Write 0.45 as 0.450. The tenths match, and in the hundredths 5 beats 0, so 0.45 is greater.
Worked example. Put 0.405, 3/8, 0.45 and 2/5 in order from least to greatest.
- Fractions to decimals: 3/8 = 3 ÷ 8 = 0.375, and 2/5 = 4/10 = 0.4.
- Three places each: 0.405, 0.375, 0.450, 0.400.
- Compare: 0.375 < 0.400 < 0.405 < 0.450, so the order is 3/8, 2/5, 0.405, 0.45.
Benchmarks save dividing. 3/8 is less than ½ (half of 8 is 4) and 5/9 is more (half of 9 is 4.5), so 3/8 < 5/9.
What costs marks
The idea: Most of these come from looking at the digits and forgetting what place they are in.
- Thinking more digits means bigger. 0.405 < 0.45.
- Testing 8 with two digits. 2364 is not divisible by 8, even though 64 is.
- Testing 6 with the digit sum. Test for 2 and for 3 instead.
- Saying 7 ÷ 0 = 0. Dividing by zero has no answer.