Calculating with integers, decimals and negatives

$500 + 145 = 645$

Negative numbers are smaller than positive ones, and $-3$ is further below zero than $-1$.

So the order is $-3,\ -1,\ 2,\ 4,\ 7$.

In 3721 the 7 is in the hundreds column.

So it is worth $7 \times 100 = 700$.

The most negative number is the smallest.

$-7 < -3 < 0 < 4 < 6$

In 62 837 the digits are: 6 ten-thousands, 2 thousands, 8 hundreds, 3 tens, 7 units.

So the 3 is worth 3 tens, i.e. $30$.

$120 - 89$: take away 90 to get 30, then add back 1.

$120 - 89 = 31$

Compare the whole-number parts first: only 0.5 starts with 0.

So the smallest is $0.5$.

Start at $-9$ on a number line and move 5 to the right.

$-9 + 5 = -4$

Pay per hour $=$ total pay $\div$ hours

$50 \div 4 = 12.5$, so £12.50 per hour.

On a number line, numbers less than $-5$ are further to the left.

So $-6$, $-7$ or $-10$ all work.

In 27.963 the first digit after the decimal point is the tenths digit.

So the 9 is worth $\frac{9}{10} = 0.9$.

Three thousand = 3 in the thousands column, one hundred = 1 in the hundreds, no tens so 0, and 7 units.

So the number is $3107$.

The negatives are smallest, and $-5$ is further below zero than $-2$.

Then the positives in order: $-5, -2, 3, 7, 9$

Brackets first: $3 + 2 = 5$.

Then $20 \div 5 = 4$.

16 007 has 1 ten-thousand, 6 thousands, 0 hundreds, 0 tens and 7 ones.

So the 6 is worth $6000$.

Amount spent: $20 - 6 = 14$ pounds

Each candle costs £2: $14 \div 2 = 7$

Simon buys 7 candles.

Negative numbers get smaller the further they are from zero, so $-7$ is the smallest.

Order: $-7, -2, -1, 0, 7$

To be even, the last digit must be 4 or 6.

The smallest start uses 3 first. Ending in 6 leaves 3, 4, 7: 3476. Ending in 4 leaves 3, 6, 7: 3674.

Smallest is 3476.

The most negative number is the smallest.

Negatives: $-11 < -7 < -2$, then positives: $3 < 8 < 10$.

Order: $-11, -7, -2, 3, 8, 10$

In 12 345 the digits are: 1 ten-thousand, 2 thousands, 3 hundreds, 4 tens, 5 ones.

So the 2 is worth 2000.

Apples: $86 + 75 + 92 = 253$

Oranges: $68 + 80 + 76 = 224$

Difference: $253 - 224 = 29$

The most negative number is the smallest.

Negatives: $-10 < -7 < -2$, then $0$, then $1 < 8$.

Order: $-10, -7, -2, 0, 1, 8$

Share equally means divide.

$£42 \div 3 = £14$

Compare whole-number parts first: 0.12 and 0.21 are less than 1.

Then compare tenths: $0.12 < 0.21$ and $1.02 < 1.20$.

Order: 0.12, 0.21, 1.02, 1.20

Give every number three decimal places: 0.320, 0.400, 0.350, 0.309.

Now compare: 0.309 < 0.320 < 0.350 < 0.400, so the order is 0.309, 0.32, 0.35, 0.4.

In 8765 the digits are 8 thousands, 7 hundreds, 6 tens and 5 ones.

So the 7 is worth 700.

The negative number furthest from zero is smallest.

$-6, -4, -3, 0, 1, 2, 7$

The digits of 1074 are thousands, hundreds, tens, ones: 1, 0, 7, 4.

The 7 is in the tens column, so it is worth $7 \times 10 = 70$.

Multiplying by 100 moves every digit two places to the left.

$31.7 \times 100 = 3170$

Work out the bracket first: $3 + 5 = 8$

Then $10 \times 8 = 80$

Add the numbers and halve: $(7 + 15) \div 2 = 22 \div 2 = 11$

The most negative number is the smallest.

Order: $-7, -4, -2, 1, 8$

Negative numbers are smaller than zero, and $-3$ is further below zero than $-1$.

So the order is $-3, -1, 0, 2, 4$

One million is $1\,000\,000$ (six zeros).

Two million is $2\,000\,000$

Green: $26 + 7 = 33$

Red: $16 + 11 = 27$

Difference: $33 - 27 = 6$

Compare the tenths digit first: 0.02, 0.152, 0.2, 0.37, 0.4

So the order is 0.02, 0.152, 0.2, 0.37, 0.4

(a) Multiply first: $3 \times 5 = 15$, then $15 + 7 = 22$

(b) $2^3 = 2 \times 2 \times 2 = 8$

(c) $35 = 7 \times 5$ and $2 + 3 = 5$, so $7 \times (2 + 3) = 35$

(a) Negatives come first, and $-6$ is smaller than $-5$: $-6, -5, 0, 6, 12$

(b) Write each with three decimal places: 0.078, 0.780, 0.870, 0.708.

Order: $0.078, 0.708, 0.78, 0.87$

In 542.3 the digits are 5 hundreds, 4 tens, 2 ones and 3 tenths.

So the 4 is worth 40.

The thousands digit is the fourth digit from the right in a whole number.

For example, $564\,000$: the digits are 5 (hundred thousands), 6 (ten thousands), 4 (thousands), then zeros.

(i) $100 - 56.3 = 43.7$

(ii) One whole is $\frac{7}{7}$, so $\frac{7}{7} - \frac{2}{7} = \frac{5}{7}$

Becky has $6 \times 2 = 12$ cousins.

David has $12 \times 2 = 24$ cousins.

In 16.35 the digit after the decimal point is the tenths digit.

So the 3 is worth 3 tenths, which is $\frac{3}{10} = 0.3$

Work out the top and bottom first: $30 + 12 = 42$ and $5 + 3 = 8$.

$42 \div 8 = 5.25$

Hire: $10 \times £45 = £450$

Total cost: $£450 + £30 = £480$

Paid: $10 \times £50 = £500$

Change: $£500 - £480 = £20$

Multiplication comes before subtraction (BIDMAS).

So $5 - 3 \times 4 = 5 - 12 = -7$, not $2 \times 4 = 8$.

Write each with two decimal places: 0.21, 0.20, 0.03, 0.10, 0.16.

Order: $0.03,\ 0.1,\ 0.16,\ 0.2,\ 0.21$.

Give each number three decimal places: 0.570, 0.507, 0.500, 0.050.

Smallest first: 0.05, 0.5, 0.507, 0.57

Add the two numbers and halve: $\frac{-2 + 8}{2} = \frac{6}{2} = 3$

(a) Paint: $2 \times 24.95 = £49.90$

Planks: $232.50 \div 7.75 = 30$

Total: $119.96 + 49.90 + 232.50 + 14.95 = £417.31$

(b) $892.48 + 4.47 + 240 - 365.50 = £771.45$

Convert to decimals: $\frac12 = 0.5$, $\frac23 \approx 0.67$, $\frac14 = 0.25$.

Smallest first: $\frac14,\ \frac12,\ \frac23$

Footballs: $5 \times 9.50 = 47.50$

Hockey sticks: 6 sticks is 3 lots of 2, so $3 \times 30 = 90$

Cricket bats: $2 \times 23 = 46$

Tennis balls: 4 balls cost $5$

Total $= 47.50 + 90 + 46 + 5 = £188.50$, which is less than £200, so he can buy everything.

Total lengths needed: $8050 \div 25 = 322$

Lengths still to swim: $322 - 178 = 144$

Tickets: $33 + 2 \times 24.50 = 82$

Meals: $15 + 2 \times 10 = 35$

Total: $23 + 82 + 35 = £140$

Left: $200 - 140 = £60$

(a) $23 \div 4 = 5.75$, so he can afford 5 whole jars (costing £20).

(b) At half price each jar is £2. $23 \div 2 = 11.5$, so he can buy 11 jars.

Twice 5 is 10, so he is not correct: he can buy 11.

Start at $-15$ and go up 42 degrees.

$-15 + 42 = 27$

Highest temperature: $27^\circ$C

Amount left after the deposit: $1428 - 150 = 1278$

Each payment: $1278 \div 6 = 213$

Each monthly payment is £213.

(i) Same denominator, and $5 > 2$, so $\frac{5}{8} > \frac{2}{8}$.

(ii) $-2 \times -3 = 6$ and $-3 + 9 = 6$, so they are equal.

Give each number three decimal places: 0.410, 0.500, 0.460, 0.408.

Now compare: 0.408 < 0.410 < 0.460 < 0.500, so the order is 0.408, 0.41, 0.46, 0.5

Nails needed: $35 \times 4 = 140$

Nails available: $3 \times 48 = 144$

$144 > 140$, so yes, she has enough (4 to spare).

Cheese: $2 \times £2.30 = £4.60$

Total: $£1.20 + £0.70 + £4.60 = £6.50$

Change: $£10 - £6.50 = £3.50$

Danny should get £3.50, so he is not correct.

Monday: $5$ °C

Tuesday: $5 - 10 = -5$ °C

Wednesday: $-5 + 3 = -2$ °C

Difference: $5 - (-2) = 7$ °C

Flour in A and B together: $2500 - 940 = 1560$ g.

They are equal, so bag A has $1560 \div 2 = 780$ g.

Multiplication comes before addition (BIDMAS).

$3 \times 4 = 12$, then $2 + 12 = 14$, not 20.

Planes: $2 \times 600 = 1200$

Hotel: $7 \times 120 = 840$

Theme park: $2 \times 250 = 500$

Total: $1200 + 840 + 500 = 2540$ dollars, and $2540 > 2500$.

Boxes needed: $200 \div 25 = 8$

Cost: $8 \times £9.75 = £78$

(a) $2 \times 5 = 10$, and $300 \div 10 = 30$

(b) $3.5 \times 12 = 42$

(c) The reciprocal of 20 is $\frac{1}{20} = 0.05$

Split 15 into 10 and 5:

$23 \times 10 = 230$

$23 \times 5 = 115$

$230 + 115 = 345$

Hotel for one person: $7 \times £50 = £350$

Cost for one person: $£150 + £350 + £250 = £750$

For 4 people: $4 \times £750 = £3000$

Seats: $14 \times 15 = 210$

Tickets sold: $1274 \div 6.50 = 196$

Not sold: $210 - 196 = 14$

Two coffees: $2 \times 2.75 = 5.50$

Total spent: $5.50 + 2.90 = 8.40$, so £8.40

Change: $10 - 8.40 = 1.60$, so £1.60

£1.60 is more than £1.50, so Dave is correct.

$14 < 21$

$4 + 7 = 11$ and $103 - 92 = 11$, so they are equal.

$2^2 = 4$ and $2 \times 2 = 4$, so they are equal.

$-3$ is nearer zero than $-5$, so $-3 > -5$

Total $= 1280 + 640 + 220 = £2140$

Each share $= 2140 \div 4 = £535$

Left after two pieces: $240 - 2 \times 45 = 150$ cm

$150 \div 40 = 3.75$

Only whole pieces count, so 3.

Books: $45 \times 1.20 = £54$

Candles: $34 \times 1.50 = £51$

Left for calendars: $150 - 54 - 51 = £45$

90p is £0.90, so $45 \div 0.90 = 50$ calendars.

(a) Use the three largest digits, biggest first: $974$.

(b) Put the two smallest digits, 1 and 2, in the tens places. Put 6 and 8 in the units places.

$16 + 28 = 44$ (or $18 + 26 = 44$), the smallest possible total.

(a) Multiplication comes before subtraction: $12 - 2 \times 4 = 12 - 8 = 4$. Jenny subtracted first.

(b) The range is the largest value minus the smallest: $8 - 1 = 7$. Rehan used the last and first numbers in the list.

Split 63 into 60 and 3.

$247 \times 60 = 14820$

$247 \times 3 = 741$

$14820 + 741 = 15561$

Cakes: $4 \times £1.30 = £5.20$

Spent: $£10 - £1.80 = £8.20$

Two tins: $£8.20 - £5.20 = £3.00$

One tin: $£3.00 \div 2 = £1.50$

$273 \times 50 = 13\,650$

$273 \times 4 = 1\,092$

$13\,650 + 1\,092 = 14\,742$

Multiply both numbers by 10: $8184 \div 12$

$81 \div 12 = 6$ r 9, $98 \div 12 = 8$ r 2, $24 \div 12 = 2$

So the answer is $682$.

(a) Negative ÷ negative is positive: $-12 \div -4 = 3$

(b) $2^5 = 2\times2\times2\times2\times2 = 32$

(c) $30 \div (3+2) - 4 = 30 \div 5 - 4 = 6 - 4 = 2$

Red: $52 \div 5 = 10.4$, so enough for 10 bracelets.

White: $80 \div 7 = 11.4...$, so enough for 11.

The red beads run out first, so the greatest number is 10.

Top: $208.3 - 15.7 = 192.6$

Bottom: $5.694 + 1.8^2 = 5.694 + 3.24 = 8.934$

$\sqrt{192.6 \div 8.934} = \sqrt{21.558...} = 4.643069317...$

Miles driven $= 47879 - 47241 = 638$

Mileage pay $= 638 \times 0.47 = £299.86$

Total $= 299.86 + 80 = £379.86$

Top: $208.3 - 15.7 = 192.6$

Bottom: $5.694 + 1.8^2 = 5.694 + 3.24 = 8.934$

$192.6 \div 8.934 = 21.5580…$

$\sqrt{21.5580…} = 4.643069317$

(a) The 07 46 reaches Salford at 08 05: that is 14 + 5 = 19 minutes.

(b)(i) He must reach Manchester by $09\,00 - 8$ min $= 08\,52$. The 08 17 from Lostock arrives at 08 45. Leave home 6 minutes earlier: 08 11.

(ii) The next Lostock train (09 17) reaches Manchester at 09 45, after 09 15, so he still needs the 08 17. No effect.

Weekdays: $5 \times 6 = 30$ hours, $30 \times 8 = £240$

Weekend rate: $1.5 \times 8 = £12$ per hour

Weekend: $(4 + 3) \times 12 = £84$

Total: $240 + 84 = £324$

Aisha's 18th birthday: $1993 + 18 = 2011$.

Elections every 5 years: 2011, 2016, 2021, 2026, 2031.

2030 is not in the list, so No, there will not be an election in 2030.

Ski hire: $5 \times 26 = £130$

Lift pass: $4 \times 45 = £180$

Lessons: $8 \times 23.50 = £188$

Total: $130 + 180 + 188 = £498$, and $498 < 500$, so Lucia has enough money.

Large plates: $6 \times £2 = £12$

Each small plate is half price: $80\text{p} \div 2 = 40\text{p}$, so $6 \times 40\text{p} = £2.40$

Total $= £12 + £2.40 = £14.40$

$£14.40 < £15$, so yes, he has enough (with 60p left over).

(a) Top: $9.8 + 6.8 = 16.6$. Bottom: $4.2 \times 2.1 = 8.82$

$16.6 \div 8.82 = 1.882086168\ldots$

(b) The third decimal place is 2, so round down: $1.88$

(a) Top: $25 - \sqrt{43.87} = 18.3765...$

Bottom: $6 + 2.1^2 = 10.41$

$18.3765... \div 10.41 = 1.765279230...$

(b) Reciprocal $= 1 \div 0.625 = 1.6$

Units used: $89\,198 - 88\,738 = 460$ kWh

Cost: $460 \times 16 = 4600 + 2760 = 7360$p

$7360\text{p} = £73.60$

Distance: $12845 - 12468 = 377$ miles

Cost: $377 \times 13 = 4901$p

$4901 \div 100 = £49.01$

Top: $2.75 \times 14.6 = 40.15$

Bottom: $10 - 1.97 = 8.03$

$40.15 \div 8.03 = 5$

(a) $367 \times 42 = 14680 + 734 = 15414$. There are 3 decimal places in total, so $3.67 \times 4.2 = 15.414$

(b) Multiply both by 10: $598.4 \div 16$

$598.4 \div 16 = 37.4$ (since $16 \times 37 = 592$ and $6.4 \div 16 = 0.4$)

$300 \div 4.85 = 61.85...$

She cannot buy part of a book, and 62 books would cost more than £300, so round down.

She can buy 61 books.

$4275 \div 28 = 152.67\ldots$, so 152 full bags.

$152 \times 28 = 4256$ sweets used.

$4275 - 4256 = 19$ sweets left.

Write each with denominator 12:

$\frac{1}{3} = \frac{4}{12}$, $\frac{3}{4} = \frac{9}{12}$, $\frac{1}{4} = \frac{3}{12}$, $\frac{7}{12}$, $\frac{1}{2} = \frac{6}{12}$

Order: $\frac{3}{12}, \frac{4}{12}, \frac{6}{12}, \frac{7}{12}, \frac{9}{12}$, i.e. $\frac{1}{4}, \frac{1}{3}, \frac{1}{2}, \frac{7}{12}, \frac{3}{4}$

Grid method:

$70 \times 50 = 3500$, $70 \times 8 = 560$, $4 \times 50 = 200$, $4 \times 8 = 32$

$3500 + 560 + 200 + 32 = 4292$

Small boxes: $3 \times 450 = 1350$ g

Left for large boxes: $5850 - 1350 = 4500$ g

$4500 \div 750 = 6$ large boxes

(a) Difference per parcel: $35.38 - 15.25 = 20.13$

$12 \times 20.13 = £241.56$

(b) He used 20 parcels instead of 21 and £15 instead of £15.25. Both are smaller than the real values, so the real cost is more than £300.