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Worked Solutions – Test 9 2021-02-05T14:14:12+00:00

## Worked Solutions – Test 9

#### Question 1 How many drinks in total were sold at this nightclub between the end of July and the end of October?

A: 11,900

B: 15,600

C: 17,900

D: 21,560

#### Written Solutions

This question is more complicated than it seems since this is a cumulative frequency graph.

By the end of October, $6800 + 6000 + 5100 = 17,900$ drinks had been sold.

By the end of July, $2800 + 2000 + 1200 = 6000$ drinks had been sold.

Since the drinks that were sold up to the end of July are included in the drinks that had been sold up to the end of October, we need to deduct the 6000 from the 17,900, so $17,900 - 6000 = 11,900$ drinks sold.

(You might prefer to forget that these figures are in hundreds for most of the calculations, provided you remember to multiply your final answer by 100.)

#### Question 2 To one decimal place, what was the percentage increase in the number of drinks sold between the end of July and the end of August?

A: 6.2%

B: 8.4%

C: 9.7%

D: 11.7%

#### Written Solutions

By the end of July, $2800 + 2000 + 1200 = 6000$ drinks had been sold.

By the end of August, $4900 + 4200 + 3600 = 12,700$ drinks had been sold.

Since the drinks that were sold up to the end of July are included in the drinks that had been sold up to the end of August, we need to deduct the 6000 from the 12,700, so $12,700 - 6000 = 6,700$ drinks sold.

Therefore between the end of July and the end of August, drink sales had increased from 6000 to 6700. As a percentage increase, this can be calculated as follows:

$\dfrac{6700-6000}{6000}\times 100 = 11.666\%$ or $11.7\%$ to one decimal place.

#### Question 3 From October sales figures, sales of all three beer brands further drop off in the month of November:  San Miguel drops by one third, Carlsberg by 12.5% and Amstel by 40%  How many beers were sold in total during the month of November?

A: 1885

B: 2215

C: 2275

D: 3642

#### Written Solutions

In October, the sales of Carlsberg were $6800 - 5800 = 1000$. If Carlsberg fell by 12.5%, then sales in November can be calculated as follows:

$1000 \times 0.875 = 875$

In October, the sales of San Miguel were $6000 - 4800 = 1200$. If this fell by $\frac{1}{3}$, then the sales in November would be:

$1200 – (1200 \times \frac{1}{3}) = 800$

or $1200 \times \frac{2}{3} = 800$

In October, the sales of Amstel were $5100 - 4100 = 1000$. If this fell by 40%, then the sales in November would be:

$1000 \times 0.6 = 600$

Therefore total beer sales in November were $875 + 800 + 600 = 2275$

#### Question 4 By the end of October, the bar had generated on overall total of €68,915 from sales of the three beer brands.  How much was the average drink sold for in pounds if the exchange rate was €1.18 to the pound?

A: £2.14

B: £2.97

C: £3.26

D: £3.82

#### Written Solutions

By the end of October, a total of $68 + 60 + 51 = 179$ hundred drinks had been sold.

Let’s convert this into figures:

$179 \times 100 = 17,900 \text{ drinks}$

If 17,900 drinks were sold for €68,915, then we can work out the average price per drink as follows:

$€68,915 \div 17,900 = €3.85 \text{ per drink}$

All we need to do now is convert this amount into pounds:

$€3.85 \div €1.18 = £3.26$

#### Question 5 To the nearest whole number, what was the difference in percentages between the percentage increase in wages between 2006 to 2008 and 2008 to 2010?

A: 2 percentage points

B: 6 percentage points

C: 9 percentage points

D: 11 percentage points

#### Written Solutions

In 2006, wages were £183,000 which increased to £212,000 in 2008. As a percentage increase, this can be calculated as follows:

$\dfrac{212-183}{183} \times 100 = 15.84\%$

In 2008, wages were £212,000 which increased to £241,000 in 2010. As a percentage increase, this can be calculated as follows:

$\dfrac{241-212}{212} \times 100 = 13.67\%$

The difference in percentages is $15.84\% - 13.678\% = 2\%$ to the nearest whole number.

(In this question, it is easier to overlook the fact that these figures are in thousands since this is a question regarding proportionality and percentages.)

#### Question 6 Between 2006 and 2010, what was the proportion of agent’s fees as a percentage of wages correct to two decimal places?

A: 9.11%

B: 11.76%

C: 14.68%

D: 16.63%

#### Written Solutions

Again, in this question, it is easier to overlook the fact that these figures are in thousands since this is a question regarding proportionality and percentages.

Wages between 2006 and 2010 came to a total of $£183 + £196 + £212 + £210 + £241 = £1042$

In the same period, agents’ fees were $£25 + £27 + £31 + £34 + £36 = £153$

Agents’ fees, as a percentage of wages, can be calculated as follows:

$\dfrac{153}{1042} \times 100 = 14.68\%$ to two decimal places.

#### Question 7 Council tax represents 55% of utilities costs.  What is the overall percentage of council tax of all costs combined?

A: 0.92%

B: 1.07%

C: 1.18%

D: 2.25%

#### Written Solutions

This question requires quite a lot of adding! Probably the most useful starting point is to add up the total utility costs over the 5 year period:$12 + 14 + 15 + 19 + 21 = 81$

(Again, it is probably easier to overlook the fact that these numbers are in thousands since this is a question about proportionality and percentages.)

Now we need to calculate the total costs. Since we have already just calculated the total of the utility costs over the 5 year period, we can use this figure rather than have to type in the same 5 utility costs again. Total fees are:

$81\text{ (from above answer)} + 183 + 196 + 212 + 210 + 241 + 416 + 416 + 422 + 428 + 432 + 25 + 27 + 31 + 34 + 36 + 67 + 71 + 73 + 79 + 84 = 3764$

(You may prefer to add up each row at a time to ensure accuracy and then add up each line total.)

If council tax is 55% of the utility costs, then the utility costs are:

$0.55 \times 81 = 44.55$

The utility costs as a percentage of total costs is:

$\dfrac{44.55}{3764} \times 100 = 1.18\%$

#### Question 8 If the theatre company had been able to save 25% of agent’s fees between 2006 to 2010 and invest the money saved into marketing, what would have been the total spent on marketing in this period?

A: £265,918

B: £412,250

C: £486,590

D: £490,267

#### Written Solutions

First of all, we need to work out what the total agents’ fees were between 2006 and 2010:

$£25 + £27 + £31 + £34 + £36 = £153$

Again, we can overlook the fact that these figures should be thousands, but we can convert this figure to thousands later.

If the theatre company had been able to save 25%, then they would have saved:

$£153 \times 0.25 (\text{or} £153 \div4) = £38.25$

The total marketing spend was $£67 + £71 + £73 + £79 + £84 = £374$

If the money they had saved on agent’s fees (£38.25) had been invested in marketing instead, then the total marketing spend would have been $£38.25 + £37 = £412.25$. Remember that this figure needs to be multiplied by 1000, so the final answer is $£412,250$

#### Question 9 From 2019 to 2020, cod catches increased by 12%, salmon by 25,000 metric tonnes and haddock by a quarter.  To the nearest whole number, what is the percentage increase in the total catch of all three fish combined from 2019 to 2020?

A: 16%

B: 20%

C: 23%

D: 27%

#### Written Solutions

In 2019, 130 thousand metric tonnes of cod were caught. If this increased by 12%, then the amount of cod caught in 2020 was:

$130 \times 1.12 = 145.6$

(Easier to ignore the fact that the numbers are in thousands since this is a question about proportionality and percentages.)

If salmon increased by 25,000 metric tonnes, then in 2020 there would have been $110 + 25 = 135$ thousand metric tonnes.

If haddock increased by a quarter from 2019, then the amount of haddock caught was:

$120 \times 1.25 (\text{or} 120 \times \frac{5}{4}) = 150 \text{ metric tonnes.}$

Therefore in 2020, a total of $145.6 + 135 + 150 = 430.6$ thousand metric tonnes of fish were caught compared to $130 + 120 + 110 = 360$ thousand in 2019.

As a percentage increase, this can be calculated as follows:

$\dfrac{430.6-360}{360} = 19.61\%$ or $20\%$ to the nearest whole number.

#### Question 10 To the nearest million, what is the value of the overall catch of cod in 2019?

A: £145 million

B: £196 million

C: £265 million

D: £283 million

#### Written Solutions

In 2019, 130 thousand metric tonnes of cod were caught. Since the price of fish is in stones, we need to convert metric tonnes to stones. First of all, it makes sense to convert 130 thousand metric tonnes to 130,000 metric tonnes. We can now convert this into stones:

$130,000 \times 157.5 = 20,475,000 \text{ stones.}$

If cod costs £13.80 per stone, then the value of the code catch in 2019 is:

$20,475,000 \times £13.80 = £282,555,000$ which is $£283$ million to the nearest million.

#### Question 11 In 2021, the price of salmon increased by 15% from 2019, but the quantity caught fell by 18% from 2019.  What was the value of the salmon catch in 2021?

A: £286 million

B: £307 million

C: £322 million

D: £415 million

#### Written Solutions

If the price of salmon increased by 15% in 2020, then it went up from £19.70 by 15%. We can work out the new price as follows:

$£19.70 \times 1.15 = £22.65$ (you may prefer to stick with the given answer of 26.555)

If the quantity of salmon fell by 18%, and was 110,000 to begin with, then the amount caught in 2020 was:

$110,000 \times 0.82 = 90,200 \text{ metric tonnes}$

Now we need to convert this to stones since the price of salmon is per stone:

$90,2000 \times 157.5 = 14,206,500 \text{ stones}$

At the increased price of £26.555 per stone, the value of the 2020 salmon catch is:

$14,206,500 \times £26.555 = £321,848,257.50$ or $£322$ million to the nearest million.

(Even if you round £26.555 to £26.56, this will not affect the final answer.)

#### Question 12 In 2019, haddock and mackerel are caught in a ratio of 8 : 3.  If the total value of the mackerel caught is £88,593,750, how much does a stone of mackerel cost?

A: £9.95

B: £11.10

C: £12.50

D: £13.65

#### Written Solutions

If haddock and mackerel are caught in a ratio of $8 : 3$, then this means that $\frac{8}{11}$ of the catch is haddock and $\frac{3}{11}$ is mackerel. (We are dealing with elevenths because the sum of the ratio is 11.)

If $\frac{8}{11} = 120,000$ metric tonnes, then we need to work out what $\frac{3}{11}$ is.

$120,000 \div8$ tells us what $\frac{1}{11}$ is worth, so $\frac{1}{11} = 15,000$

Therefore $\frac{3}{11} = 15,000 \times 3 = 45,000$

So 45,000 metric tonnes of mackerel are caught with a value of £88,593,750.

The price per metric tonne is therefore:

$£88,593,750 \div 45,000 = £1968.75$

If this is the price per metric tonne, then the price per stone is:

$£1968.75 \div 157.5 = £12.50$

#### Question 13  What was the mean annual percentage of TecSmart’s sales were generated by the Liverpool office?

A: 17.9%

B: 18.1%

C: 18.7%

D: 19.1%

#### Written Solutions

(We can overlook the fact that these figures should be thousands since this is a question about percentages and proportionality.)

In 2016, the total sales of Tec Smart was $£316 + £185 + £192 + £212 + £206 = £1111$. The Liverpool office generated 206 of these sales, so as a percentage this can be calculated as follows:

$£206 \div £1111 \times 100 = 18.54\%$

(Since the answers are given to one decimal place, I would suggest rounding this value to one decimal place as well, so we will go with $18.5\%$)

In 2017, total sales were $£349 + £202 + £208 + £219 + £228 = £1206$. The Liverpool office generated £228 of these sales, so as a percentage this can be calculated as follows:

$£228 \div £1206 \times 100 = 18.90%$

Therefore the mean percentage increase is ($15.5% + 15.9%) \div 2 = 18.7\%$

#### Question 14  Which city showed the greatest percentage increase in sales between 2016 and 2017?

A: London

B: Bristol

C: Birmingham

D: Liverpool

#### Written Solutions

In 2016, the London office made 316 sales and made 349 in 2017.

As a percentage increase, this can be calculated as follows:

$\dfrac{349-316}{316} \times 100 = 10.44\%$

In 2016, the Bristol office made 316 sales and made 349 in 2017.

As a percentage increase, this can be calculated as follows:

$\dfrac{202-185}{185} \times 100 = 9.18\%$

In 2016, the Birmingham office made 316 sales and made 349 in 2017.

As a percentage increase, this can be calculated as follows:

$\dfrac{208-192}{192} \times 100 = 8.33\%$

Note that Manchester doesn’t feature as an answer option, so don’t waste time calculating the figures for this!

In 2016, the Liverpool office made 316 sales and made 349 in 2017.

As a percentage increase, this can be calculated as follows:

$\dfrac{228-206}{206} \times 100 = 10.67\%$

Therefore, Liverpool was the office which had the biggest percentage increase in sales.

#### Question 15  In 2018, the London branch’s sales were down from the previous year by 10%, but the Bristol branch was up by 14%, the Birmingham branch up by 12.5% and Manchester and Liverpool both up by 11%.  How much profit was made given that profit is 42% of sales?

A: £462,354

B: £535,311

C: £664,936

D: £844,109

#### Written Solutions

If the London sales were down 10% from 2017, then the total sales in 2018 can be calculated as follows:

$£349 \times 0.9 = £314.1$

If the Bristol sales were up 14% from 2017, then the total sales in 2018 can be calculated as follows:

$£202 \times 1.14 = £230.28$

If the Birmingham sales were up 12.5% from 2017, then the total sales in 2018 can be calculated as follows:

$£208 \times 1.125 = £234$

(Again, we can overlook the fact that the figures should be in thousands and this will not affect the final answer.)

If both the Manchester and Liverpool offices were up by 11%, then it is much easier to work this out as one calculation rather than doing them separately. Together, these offices generated $£219 + £228 = £447$. Therefore, their combined total sales in 2018 can be calculated as follows:

$£447 \times 1.11 = £496.17$

Therefore total sales came to a total of $£496.17 + £234 + £230.28 + £314.1 = £1274.55$

If 42% is profit, then the total profit made can be calculated as follows:

$£1274.55 \times 0.42 = £535.311$

So far we have overlooked the fact that these figures are in thousands, so for the final answer we simply need to multiply £535.311 by 1,000.$£535.311 \times 100 = £535,311$

#### Question 16  If the Birmingham and Liverpool branches’ combined sales in 2017 are made by either Visa or Mastercard in the ratio of 5:3 what was the total value of Mastercard sales?

A: £128,450

B: £163,500

C: £170,005

D: £181,980

#### Written Solutions

If the Birmingham and Liverpool branches’ sales are made by Visa and Mastercard in a ratio of $5 : 3$, this means that $\frac{5}{8}$ of the sales are through Visa and $\frac{3}{8}$ by Mastercard. (We are dealing with eights here because the sum of the ratio is 8.)

The total sales of the Birmingham and Liverpool offices in 2007 are $£208 + £228 = £436$

(Again, it is probably easier at this stage to overlook the fact that these figures are in thousands for the sake of ease.)

If total sales were £435 and $\frac{3}{8}$ were made by Mastercard, then the value of Mastercard sales can be calculated as follows:

$£436 \times \frac{3}{8} = £163.50$

We need to remember that we have overlooked the fact that the figures are in thousands, so we need to multiply this answer by 1000:

$£163.50 \times 1000 = £163,500$

#### Question 17 To one decimal place, what was the mean percentage increase in sales between August to December for the 3 oven companies?

A: 6.2%

B: 7.3%

C: 8.9%

D: 11.6%

#### Written Solutions

(Since this is a question about percentages and proportionality, it is more convenient to overlook the fact that the figures are in hundreds.)

Company A’s sales decreased from 25 to 18. As a percentage decrease, this can be calculated as follows:

$\dfrac{25-18}{25} \times 100 = 28\%$

Company B’s sales increased from 40 to 46. As a percentage increase, this can be calculated as follows:

$\dfrac{46-40}{40} \times 100 = 15\%$

Company C’s sales increased from 20 to 27. As a percentage increase, this can be calculated as follows:

$\dfrac{27-20}{20} \times 100 = 35\%$

Therefore the mean percentage increase is:

$15\% - 28\% + 35\% = 7.3\%$ to one decimal place.

(Since Company B was a percentage decrease rather than a percentage increase, this should be considered as a negative percentage increase which is why we are subtracting it in the above calculation.)

#### Question 18 If the percentage increase in combined sales from December to January is the same as the percentage increase of combined sales from September to October, how much revenue will be generated in January if the mean oven sale generates £415?  Give your answer to the nearest £10,000.

A: £3,010,000

B: £4,200,000

C: £4,600,000

D: £5,400,000

#### Written Solutions

Combined sales in September were $47 + 34 + 29 = 110$

Combined sales in October were $56 + 41 + 37 = 134$

If sales increased from 110 to 134, as a percentage increase, this can be calculated as follows:

$\dfrac{134-110}{110} \times 100 = 21.81\%$

We are told that the percentage increase in sales from December to January is the same as September to October, so sales increased by 21.81% from December to January.

The total sales in December were $46 + 27 + 18 = 91$

Remember that these are hundreds of sales so the total number of sales is $91 \times 100 = 9100$

If sales increased by 21.81% from December to January, then in January there would be $1.2181 \times 9100 = 11084.71$ sales

If the average sale price of an oven was £415, then the total value of sales in January was £415 $\times 11084.71 = £4,600,154.65$ which would be $£4,600,000$ to the nearest ten thousand.

(You can round the sales to 11,085 and the answer will remain unaffected if you are not comfortable with 0.71 of a sale.)

#### Question 19 In November £4,836,000 is generated from combined sales.  What is the mean sale price of an oven in Australian dollars to the nearest hundred dollars?

A: AU$800 B: AU$900

C: AU$1000 D: AU$1200

#### Written Solutions

There are two approaches to this question. We can convert the total combined sales into Australian dollars and then work out the mean sale price, or we can work out the mean sale price in pounds and then convert this mean price into dollars. Neither method is better than the other!

In November, there were $43 + 28 + 22 = 93$ sales. Remember that these figures should be in hundreds, so there are in fact 9300 sales. If the total value of all sales was £4,836,000, then the mean sale price of an oven in pounds is:

$£4,836,000 \div 9300 = £519.89$

Now we need to convert this figure into Australian dollars. By not deleting this answer from our calculator display, we not only save time, but we also can keep the entire answer which will ensure greater accuracy in the next calculation (although if you were to round this mean price to £531.43, it is unlikely to lead to a massively different answer).

To convert this amount into Australian dollars, we have to convert into US dollars first.

£519.89 in US dollars can be calculated as follows:

$£519.89 \times \1.63 = \847.42$

Now we need to convert the US dollars into Australian dollars:

$847.42\times$AU$\1.45 =$AU$\1228.77$ or AU$\1200$ to the nearest one hundred dollars.

#### Question 20 In July, the sales of the 3 companies A, B and C were shared in the ratio 7 : 6 : 8.  If 10,500 ovens were sold at an average price of £462 and 2/5 of customers bought from company B using AMEX, what was the total value of company B’s AMEX sales in July?

A: £468,900

B: £554,400

C: £684,500

D: £864,500

#### Written Solutions

If the sales of companies A, B and C are in the ratio of $7 : 6 : 8,$ then this means that company A made $\frac{7}{21}$ of total sales, company B $\frac{6}{21}$ and company C $\frac{8}{21}$

(We are dealing with twenty-oneths (if that is even a word!) since the sum of the ratio is 21.)

If 10,500 ovens were sold, we need to work out how many ovens company B sold as follows:

$10,500 \times\frac{6}{21} = 3000$ ovens sold

If the average selling price was £462, then company B’s oven sales had an overall value of $£462 \times 3000 = £1,386,000$

If $\frac{2}{5}$ of the sales were made with an Amex card, then the total amount taken by Amex can be calculated as follows:

$£1,386,000 \times\frac{2}{5} = £554,400$

(Alternatively, you could have worked out $\dfrac{2}{5}$ of the total ovens sold and then multiplied this answer by the average selling price of an oven:

$\frac{2}{5} \times 3000 \times £462 = £554,400)$