Percentage change and reverse percentages

$10\%$ of 240 is 24, so $20\%$ is $48$

$240 + 48 = 288$

Monday: $\frac{1}{10}$ of £500 is £50, so the price is $500 - 50 = £450$.

Tuesday: 20% of £450 is $0.2 \times 450 = £90$, so the price is $450 - 90 = £360$.

£360 is less than £400, so yes, Chris can buy it.

Week 1: $65 \times 0.8 = £52$

Week 2: $52 - 10 = £42$

£42 is more than £40, so Maria does not have enough.

A multiplier of 0.7 means the sale price is 70% of the normal price.

The reduction is $100\% - 70\% = 30\%$, so $R = 30$

(a) $285 \times 0.93 = £265.05$

(b) The bonus is $2100 - 1700 = £400$.

$5\% = 400$, so $1\% = 80$ and $100\% = 80 \times 100 = £8000$.

20% of the price is £240.

10% is £120, so 100% is $120 \times 10 = £1200$

Number of bags: $3000 \div 150 = 20$

Total needed with 35% profit: $17.60 \times 1.35 = 23.76$

Price per bag: $23.76 \div 20 = 1.188$

Round up so the profit is at least 35%: £1.19

Chic Decor: $15 \div 3 = 5$ lots of £36, so $5 \times 36 = £180$

Style Papers: $3 \times 70 = £210$, then $210 \times 0.88 = £184.80$

£180 < £184.80, so buy from Chic Decor.

Loss: $£80 - £56 = £24$

Percentage loss: $\frac{24}{80} \times 100 = 30\%$

Tamara: $220\,000 \times 0.8 = £176\,000$

Rahim: $160\,000 \times 1.3 = £208\,000$

$208\,000 > 176\,000$, so Rahim's house has the greater value.

Estimate: $H = 4 \times 110 + 12 = 452$ m

Difference: $452 - 442 = 10$ m

$5\%$ of 442 $= 22.1$ m

$10 < 22.1$, so the difference is less than 5% of the real height ($\frac{10}{442} \times 100 = 2.26\%$).

Profit: $158\,220 - 146\,500 = 11\,720$

Percentage profit $= \frac{11\,720}{146\,500} \times 100 = 8\%$

$5\text{ kg} = 5000\text{ g}$, so there are $5000 \div 250 = 20$ bags.

Takings: $20 \times £0.65 = £13$

Profit $= £13 - £10 = £3$

Percentage profit $= \frac{3}{10} \times 100 = 30\%$

(a) $1.3 = 130\%$, which adds 30%. A 3% increase uses $1.03$: $150 \times 1.03 = 154.5$.

(b) Decrease by 3%: $100\% - 3\% = 97\% = 0.97$

$150 \times 0.97 = 145.5$

Increase: $883 - 245 = 638$

Percentage increase: $\frac{638}{245} \times 100 = 260.4\%$

First class: $\frac{85 - 76}{76} \times 100 = 11.84\%$

Second class: $\frac{66 - 65}{65} \times 100 = 1.54\%$

$7 \times 1.54 = 10.77\%$, and $11.84 > 10.77$, so Filip is correct.

One part: $240 \div 8 = 30$, so 90 pens and 150 pencils.

Cost: $90 \times 9 + 150 \times 6 = 1710$p

Income: $90 \times 11 + 150 \times 10 = 2490$p

Profit: $2490 - 1710 = 780$p

$\frac{780}{1710} \times 100 = 45.6\%$

After a 15% reduction the sale price is 85% of the normal price.

Normal price $= 46.75 \div 0.85 = £55$

The sale price is $100\% - 40\% = 60\%$ of the normal price.

$60\%$ is £660, so $1\%$ is £11 and $100\%$ is £1100

Income: $54 \times 7.50 = £405$

Costs: $100 + 120 + 80 = £300$

Profit: $405 - 300 = £105$

Percentage profit $= \frac{105}{300} \times 100 = 35\%$

After the increase the population is 109% of the 2018 figure.

$165680 \div 1.09 = 152000$

Profit $= 135 - 120 = £15$

Percentage profit $= \frac{15}{120} \times 100 = 12.5\%$

After a 15% decrease the car is worth 85% of the original.

$0.85 \times \text{original} = 13\,600$

Original $= 13\,600 \div 0.85 = £16\,000$

The sale price is $100\% - 15\% = 85\%$ of the normal price.

$272\,000 \div 0.85 = £320\,000$

(a) £551 is 95% of the normal price: $551 \div 0.95 = 580$, so £580.

(b) After year 1: $6000 \times 1.024 = 6144$.

Years 2 and 3: $6144 \times 1.017^2 = 6354.67$.

The investment is worth £6354.67.

Managers get $0.4 \times 2100 = £840$, so salesmen get $2100 - 840 = £1260$.

Each salesman: $1260 \div 7 = £180$.

Shared equally: $2100 \div 10 = £210$.

25% more than £180 is $180 \times 1.25 = £225$.

$£210 \ne £225$, so the salesman is wrong (it is only about 16.7% more).

5 kg $= 5000$ g, so $5000 \div 250 = 20$ bags

Takings $= 20 \times 65\text{p} = £13$

Profit $= 13 - 10 = £3$

Percentage profit $= \frac{3}{10} \times 100 = 30\%$

Original pressure: $70 \div 20 = 3.5$ N/cm²

New pressure: $80 \div 30 = 2.67$ N/cm²

Decrease: $3.5 - 2.67 = 0.83$

Percentage decrease: $\frac{0.83}{3.5} \times 100 \approx 24\%$

This is more than 20%, so Helen is wrong.

Mia's sale was 90% of her price: $162\,000 \div 0.9 = 180\,000$

Jerry's sale was 120% of his price: $180\,000 \div 1.2 = 150\,000$

Jerry paid £150 000.

After a 30% reduction the sale price is 70% of the normal price.

$70\% = £280$, so $10\% = £40$ and $100\% = £400$.

$Q = 1.5P$ and $R = 1.5Q = 1.5^2 P = 2.25P$

$\frac{P}{R} = \frac{1}{2.25} = \frac{4}{9}$

Yellow $= \frac{2}{5} \times 50 = 20$ litres = 4 tins; blue $= 30$ litres = 3 tins.

Cost $= 4 \times 26 + 3 \times 48 = 104 + 144 = £248$ for 50 litres.

Cost per 10 litre tin $= 248 \div 5 = £49.60$.

Profit $= 66.96 - 49.60 = £17.36$

$\frac{17.36}{49.60} \times 100 = 35\%$

Jack: 125 g for the price of 100 g, so 1 g costs $\frac{100}{125} = 0.8$ of the old price.

Sadia needs the price of the normal box to be 80% of the original.

Reduction $= 100\% - 80\% = 20\%$