Number Review: Mixed Practice

Worked example

1. 1

   $7,12,18,21,25,27,32$ — The list. Take one word from the question at a time and test every number against it.

2. 2

   $7$ — Prime: only 1 and 7 divide it. Exactly two factors is what prime means.

3. 3

   $25=5\times 5$ — 25 is 5 times itself, so it is the square number.

4. 4

   $27=3\times 3\times 3$ — 27 is 3 times itself, times itself again, so it is the cube number.

5. 5

   $36÷12=3$ — 12 goes into 36 exactly, with nothing left over, so 12 is a factor of 36. (18 works too: 36 ÷ 18 = 2.)

6. 6

   $27=9\times 3$ — 27 is in the 9 times table, so it is a multiple of 9. (18 = 9 × 2 works too. Write just one.)

Worked example

1. 1

   $5\times 5\times 5=125$ — 5 times itself, times itself again, makes 125. That is a cube, so 125 is the cube of 5.

2. 2

   $56÷8=7$ — 8 goes into 56 exactly, 7 times. So 8 is a factor of 56.

3. 3

   $56=8\times 7$ — 56 is in the 8 times table, so 56 is a multiple of 8. The small number is the factor, the big one the multiple.

4. 4

   $6\times 6=36$ — 6 times itself makes 36, so 36 is the square of 6.

Worked example

1. 1

   $9,14,23,37,46,60$ — The list. Find two numbers with a sum of 51, then two with a difference of 32.

2. 2

   $\text{sum }51:\text{last digits make 1}$ — The two last digits must add to something ending in 1. Only 4 + 7 = 11 does, so try 14 and 37.

3. 3

   $14+37=51$ — Add to check. It makes exactly 51, so the answer is 14 and 37.

4. 4

   $\text{difference }32:14+32=46$ — Difference means bigger minus smaller. Add 32 to each number and look for the answer in the list: 14 + 32 = 46 is there.

5. 5

   $46-14=32$ — Check by subtracting: bigger minus smaller is 32. So the answer is 14 and 46.

Worked example

1. 1

   $\text{try }3\text{ and }5$ — Pick two small odd numbers. Small numbers keep the sum easy.

2. 2

   $3+5=8$ — Add them, as the claim says.

3. 3

   $8\text{ is even}$ — 8 ends in 8, so it is even, not odd. Write: “3 + 5 = 8, which is even, so Sam is wrong.”

Worked example

1. 1

   $\text{try the prime }2$ — Claims about primes often break with 2. It is prime, and it is the only even prime.

2. 2

   $2\times 3=6$ — Multiply, as the claim says.

3. 3

   $6\text{ is even}$ — 6 is not odd, so Lee is wrong. Any other prime would not work here: odd × 3 is odd.

Worked example

1. 1

   $14212÷34=418$ — The fact you are given. Use it, so you do not need a new long multiplication.

2. 2

   $418\times 34=14212$ — Turn the division round. 14 212 shared into 34 gives 418, so 34 lots of 418 make 14 212.

3. 3

   $418\times 35=418\times 34+418$ — 35 lots is 34 lots plus one more lot of 418.

4. 4

   $=14212+418$ — Swap in the fact from the line above.

5. 5

   $=14630$ — Add: 14 212 + 418 = 14 630. Writing the add is what earns the marks.

Worked example

1. 1

   $\text{cubes: }1,8,27$ — Cubes are the rarest, so list them first. 64 is too big, because the three numbers must add to less than 40.

2. 2

   $A=1:B=2$ — 2 is not a square (1 × 1 = 1, 2 × 2 = 4), so 1 fails.

3. 3

   $A=8:B=9=3\times 3$ — 9 is a square, so B = 9 passes.

4. 4

   $C=10=2\times 5$ — 10 is in the 5 times table, so C passes too.

5. 5

   $8+9+10=27<40$ — Check the last test: 27 is less than 40. So A = 8, B = 9, C = 10.

Worked example

1. 1

   $11,31,41\text{are prime}$ — Do not guess. These fail, because each has only 1 and itself as factors.

2. 2

   $21=3\times 7$ — 21 splits as 3 × 7, so it has more than two factors. It is not prime. So 21 is an answer.

Worked example

1. 1

   $3\_\_\_\_\_\_$ — “Three million” fills the millions. Two groups of three digits must follow it.

2. 2

   $3040\_\_\_$ — “Forty thousand” fills the thousands group. It has no hundred thousands, so that place gets a 0.

3. 3

   $3040605$ — “Six hundred and five” fills the last group. There are no tens, so the middle digit is 0.

4. 4

   $3040605$ — Read it back to check: three million, forty thousand, six hundred and five. It matches.

Worked example

1. 1

   $3\times 1.85=5.55$ — Cost of the pens: 3 lots of $1.85.

2. 2

   $2\times 2.40=4.80$ — Cost of the notebooks: 2 lots of $2.40.

3. 3

   $5.55+4.80=10.35$ — Add the two costs to get the whole bill: $10.35.

4. 4

   $20-10.35=9.65$ — Change is what you paid minus the bill. Lina gets $9.65.

Worked example

1. 1

   $230÷48=4.79\dots$ — Share the students into buses of 48.

2. 2

   $4\times 48=192$ — Test 4 buses: they hold only 192, so 38 students would be left behind. 4 is impossible.

3. 3

   $5\text{ buses}$ — Round up, not down. A part-full bus is still a bus: 5 × 48 = 240 seats.