Lesson Guide
Explanation
Two steps, then the arithmetic. The smallest four-digit number is one thousand — not eleven eleven and not twelve thirty-four — and the largest two-digit number is ninety-nine. Multiplying is easier as a thousand times a hundred, less a thousand: a hundred thousand take a thousand, ninety-nine thousand. With no calculator in the room, look for that round-number shortcut before starting a long multiplication.
Your Turn
What is the largest 3-digit number times the smallest 2-digit number? Check: 999 × 10 = 9 990.
Watch Out
The smallest 4-digit number is 1000. A number cannot start with 0, but the rest can all be zeros.
Explanation
A favourite on this paper: round the same number two different ways and find the difference. Four million seven hundred and sixty-two thousand eight hundred and ninety-three, to the nearest million, looks at the hundred-thousands digit — seven — and rounds up to five million. To the nearest hundred thousand, look at the ten thousands — six — so it rounds up to four million eight hundred thousand. The difference is two hundred thousand. Round twice, then subtract; never round the difference.
Your Turn
Round 3 648 000 to the nearest million and the nearest hundred thousand. Check: 4 000 000 and 3 600 000.
Watch Out
Look at the digit immediately to the right of the rounding place, not at the whole tail.
Explanation
When a question describes two sets of numbers, write both lists out before comparing anything. Multiples of five under forty run five, ten, fifteen and so on to thirty-five. Even multiples of seven are the ones that are also even — fourteen and twenty-eight — which are just the multiples of fourteen. Then read carefully which extreme is wanted: the largest of the first list minus the smallest of the second is thirty-five take fourteen, twenty-one.
Your Turn
List the even multiples of 7 under 40. Check: 14 and 28.
Watch Out
"Even multiples of 7" is not "multiples of 7". It means the multiples of 14.
Explanation
For a place-value ratio, work out what each digit is worth and divide. In thirty thousand and sixty, the three sits in the ten thousands and is worth thirty thousand; the six sits in the tens and is worth sixty. Thirty thousand divided by sixty is five hundred, so the three is worth five hundred times the six. Counting places alone is not enough here, because the digits themselves are different — do the division.
Your Turn
In 40 080, the 4 is worth how many times the 8? Check: 40 000 ÷ 80 = 500.
Watch Out
Use what the digits are WORTH, not the digits themselves. It is 30 000 ÷ 60, not 3 ÷ 6.
Explanation
The hardest items are three or four steps, and the way through is to write each step down. How many multiples of three up to a hundred are not multiples of eight? A hundred divided by three is thirty-three remainder one, so thirty-three multiples of three. A number that is a multiple of both three and eight is a multiple of twenty-four, their lowest common multiple, and there are four of those up to a hundred. Thirty-three take four is twenty-nine. Quiz and exercises now.
Your Turn
How many multiples of 3 are there up to 100? Check: 100 ÷ 3 = 33 r 1, so 33.
Watch Out
Multiples of both 3 and 8 are multiples of 24, not of 11. Use the lowest common multiple.
📌 Key Points — this lesson in a nutshell
- Every question on this paper is two to four steps. Write each step down — the marks are lost between steps, not inside them.
- Smallest and largest: the smallest 4-digit number is 1000, the largest 2-digit is 99. Read which one the question wants.
- Rounding twice: round the same number two different ways, then subtract. Do not round the difference.
- Place-value ratio: the 3 in 30 060 is worth 30 000 and the 6 is worth 60, so the 3 is worth 500 times the 6.
Step 1: Introduction
Step 1 of 5
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