Integers

Worked example

1. 1

   $6,9,15,17,27,36,40$ — The list. Test one word at a time against these numbers.

2. 2

   $17$ — Prime: no number from 2 to 16 divides 17 exactly, so its only factors are 1 and 17.

3. 3

   $27=3\times 3\times 3$ — A cube number: it is 3 times itself, times itself again.

4. 4

   $9=3\times 3$ — A square number: it is 3 times itself.

5. 5

   $30÷15=2$ — So 15 is a factor of 30: it divides 30 with nothing left over.

6. 6

   $36÷4=9,36÷6=6$ — Both divide exactly, so 36 is a multiple of both 4 and 6. It is in both times tables.

7. 7

   $40$ — Even: it ends in 0, so it divides by 2 exactly.

Worked example

1. 1

   $6\times 20=120$ — First find all the sweets: 6 bags with 20 in each.

2. 2

   $120÷26=4.6\dots$ — Divide by the number of children to see how many each could get.

3. 3

   $4\text{each}$ — Round DOWN. A child cannot get 0.6 of a sweet, and 5 each would need 130 sweets.

4. 4

   $26\times 4=104$ — Multiply back to find how many sweets are handed out.

5. 5

   $120-104=16$ — Left over is all the sweets minus the ones handed out, so 16 are left.

Worked example

1. 1

   $25÷3.60=6.94\dots$ — Divide the money by the price to see how many books fit.

2. 2

   $6\text{books}$ — Round DOWN to 6. You cannot buy 0.94 of a book, and 7 books cost $25.20, more than you have.

3. 3

   $6\times 3.60=21.60$ — Multiply back to find what the 6 books cost.

4. 4

   $25-21.60=3.40$ — The change is your money minus what you spent: $3.40. Money always gets two decimal places.

Worked example

1. 1

   $4052-1376$ — “How much bigger” means take the smaller number from the bigger one.

2. 2

   $=2676$ — Subtract in columns, units under units. Borrow when the top digit is smaller. This is the difference.

3. 3

   $2676+1376=4052$ — Check: add the difference back onto the smaller number. You get the bigger one, so it is right.

Worked example

1. 1

   $3486+1957$ — A river is 3486 km long. Another is 1957 km longer. “Longer” means add the extra on.

2. 2

   $=5443$ — Add in columns, carrying 1 whenever a column passes 9. The longer river is 5443 km.

3. 3

   $5443-3486=1957$ — Check: the gap between the two rivers is the 1957 you were given.

Worked example

1. 1

   $\square +468=915$ — 468 was added to the box.

2. 2

   $\square =915-468=447$ — Taking away undoes adding. Check: 447 + 468 = 915.

3. 3

   $\square \times 24=1680$ — This time the box was multiplied by 24.

4. 4

   $\square =1680÷24=70$ — Dividing undoes multiplying. Check: 70 × 24 = 1680.

Worked example

1. 1

   $3888÷\square =24$ — 3888 shared into the box’s number of groups gives 24.

2. 2

   $\square =3888÷24$ — So 3888 shared into 24 groups gives the box. Swap the box and the 24.

3. 3

   $\square =162$ — Check: 3888 ÷ 162 = 24, so 162 is right.

Worked example

1. 1

   $5$ — The answer at the end. Work backwards from here.

2. 2

   $5^2=25$ — The last step was a square root, so undo it by squaring.

3. 3

   $25\times 3=75$ — Before that you divided by 3, so undo it by multiplying by 3. The number is 75.

4. 4

   $75÷3=25,\sqrt{25}=5$ — Check forwards: you get 5, so 75 is right.

Worked example

1. 1

   $300-200=100$ — Find the jump between two labels that sit next to each other.

2. 2

   $100÷5=20$ — There are 5 gaps between the labels, so each gap is worth 20.

3. 3

   $200+3\times 20=260$ — The arrow is 3 gaps past 200, so add three 20s.

Worked example

1. 1

   $35-34=1$ — The jump between the two labels is 1.

2. 2

   $1÷5=0.2$ — 5 gaps share that 1, so each gap is worth 0.2.

3. 3

   $34+4\times 0.2=34.8$ — The arrow is 4 gaps past 34, so add four lots of 0.2.

4. 4

   $34.4-34=0.4$ — To mark 34.4 instead, first find how far it is past 34.

5. 5

   $0.4÷0.2=2$ — That is 2 gaps, so draw the arrow at the 2nd mark after 34.

Worked example

1. 1

   $3.8+7.4=11.2$ — Add the two numbers, lining up the decimal points.

2. 2

   $11.2÷2=5.6$ — Halve the total. 5.6 is halfway.

3. 3

   $5.6-3.8=1.8$ — Check: the distance from the smaller number up to 5.6.

4. 4

   $7.4-5.6=1.8$ — The distance from 5.6 up to the bigger number is the same, so 5.6 is exactly in the middle.

Worked example

1. 1

   $\text{largest: }8631$ — Biggest digit first. The first place (thousands) is worth the most, so it gets the 8.

2. 2

   $\text{smallest: }1368$ — Smallest digit first, then the next smallest, so each place holds as little as it can.

3. 3

   $\text{even: ends in }6\text{ or }8$ — An even number must end in an even digit.

4. 4

   $\text{largest even: }8316$ — End in 6, so the 8 is free to go first. Then 3, 1, biggest first. (6318 is smaller.)

5. 5

   $\text{odd: ends in }1\text{ or }3$ — An odd number must end in an odd digit.

6. 6

   $\text{smallest odd: }1683$ — End in 3, so the 1 is free to go first. Then 6, 8, smallest first. (3681 is bigger.)