## Worked Solutions – Test 7

**Question 1**

Using the YoY changes as a predictor, what will the percentage increase in profit for Waves by be 2021? Give your answer to 1dp.

A: 22.9%

B: 23.7%

**C: 24.5%**

D: 25.3%

**Written Solutions**

** Step 1:** First calculate the current total cost in 2018

897 - 644 = 253\text{ euros}

** Step 2:** Now increase turnover by 4.8% (a multiplier of 1.048) for each of the three years, and decrease costs by 3% (a multiplier of 0.97) for each of the three years.

We raise the multiplier to a power of 3 for each of the three years this is compounded over.

897\times1.048^3=1032.47

253\times0.97^3=230.91

Now calculate the profit in 2021

1032.47-230.91=801.56\text{ euros}

** Step 3:** Now find the percentage increase in profit from 2018 to 2021.

\dfrac{801.56-644}{644}\times100=\bold{24.5\%}

**Video Solutions**

**Question 2**

What proportion of overall non-R&D costs are represented by Goodman’s Foods and Waves in 2018 in pounds? Give your answer to 1dp.

**A: 64.6%**

B: 64.8%

C: 65.0%

D: 65.2%

**Written Solutions**

** Step 1:** Calculate the non-R&D costs for each subsidiary. First calculate overall costs by subtracting profit from turnover, then take off R&D costs.

\begin{aligned}\text{Hatchet: }&(435-356)-65=\pounds14\text{m} \\ \text{Goodman’s Foods: }&(651-301)-91=\$259\text{m} \\ \text{Sol: }&(450-250)-36=\pounds164\text{m} \\ \text{Waves: }&(897-644)-92=161\text{m euros}\end{aligned}

** Step 2:** Convert these costs into pounds by dividing by the exchange rate.

\begin{aligned}\text{Goodman’s Foods: }&259 \div 1.4 = \pounds185\text{m} \\ \text{Waves: }&161 \div 1.15 = \pounds140\text{m}\end{aligned}

** Step 3:** Now we can add up the total non-R&D costs and the total contribution from Goodman’s Foods and Waves. Finally calculate the proportion that these two represent.

14+185+ 164+140=\pounds503\text{m}

185+140=\pounds325\text{m}

\dfrac{325}{503}\times100=\bold{64.6\%}

**Video Solutions**

**Question 3**

17 from every 20 sales for Sol are currently due to a single product, Solsense. Sales for this product is declining at a rate of 4% per year. What proportion of total revenue in 2019 does Solsense account for? Give your answer to 1dp.

A: 81.9%

B: 82.2%

C: 82.5%

**D: 82.8%**

**Written Solutions**

** Step 1:** First calculate the sales for Solsense in 2018. 17 of every 20 sales is equivalent to 85%, or a multiplier of 0.85.

0.85\times450=\pounds382.5\text{m}

We know that these sales are to decrease by 2019, so use the multiplier 0.96 for a decline of 4%.

0.96\times382.5=\pounds367.2\text{m}

** Step 2:** Additionally, calculate the overall new turnover in 2019 due to the YoY decrease of 1.5% – a multiplier of 0.985.

0.985 \times450 = \pounds443.25\text{m}

** Step 3:** Finally calculate proportion for these figures.

\dfrac{367.2}{443.25}\times100 = \bold{82.8\%}

**Video Solutions**

**Question 4**

Costs for Hatchet are predicted to overtake turnover if current trends continue. By what year will this occur?

A: 2036

B: 2037

**C: 2038**

D: 2039

**Written Solutions**

** Step 1:** We need an expression for costs in a given year. First start by calculating the total costs in 2018.

435-356=\pounds79\text{m}

Costs increase by 10% each year, which is a multiplier of 1.1 – if we want to compound this over several years, say n years, we raise this to the power n.

79\times1.1^n

Which gives us a way to calculate costs in year n.

** Step 2:** As before, write a similar expression for turnover. This increases by 1% each year, or a multiplier of 1.01.

435\times1.01^n

** Step 3:** Use these expressions, substituting values for n in until costs are greater than turnover.

The value for n that first works is

**n=20.**

79\times1.1^{20} = 531.47 > 435\times1.01^{20}=530.78

N=20 corresponds to the year** 2038.**

**Video Solutions**

**Question 5**

The difference between profit and projected profit is known as a ‘residual’. What is the percentage decrease in residual from Q2 to Q4?

**A: 72.7%**

B: 71.4%

C: 70.1%

D: 68.8%

**Written Solutions**

** Step 1:** Calculate the residual for Q2 and Q4 separately.

14-3=11\text{(000)}

-9-(-12)=3\text{(000)}

** Step 2:** Calculate the percentage decrease.

\dfrac{3-11}{11}\times100=\bold{-72.7\% \equiv 72.7\%\text{ decrease}}

**Video Solutions**

**Question 6**

Express the overall costs from Q1 to Q4 as a ratio compared to predicted costs. Write your ratio in the form n:1

A: 1.52: 1

B: 1.55: 1

**C: 1.58: 1**

D: 1.61: 1

**Written Solutions**

** Step 1:** First calculate total actual costs. Subtract profit from revenue.

(29-24)+(33-3)+(24--9)+(13--12)=\pounds93\text{(000)}

** Step 2:** Calculate total projected costs.

(29-26)+(33-14)+(24-9)+(13--9)=\pounds59\text{(000)}

** Step 3:** Now write down the ratio and simplify by dividing through by the lower number.

\begin{gathered}93,000\colon59,000 \\ \dfrac{93,000}{59,000}\colon1 \\ \bold{1.58\colon1} \end{gathered}

**Video Solutions**

**Question 7**

At some point between Q2 and Q3 the company becomes unprofitable. This was predicted to occur between Q3 and Q4. Estimate the months that these occur in and state the difference between the two.

**A: 3 months**

B: 4 months

C: 5 months

D: 6 months

**Written Solutions**

** Step 1:** From Q2 to Q3 profit changes by -12(£000’s). Each quarter is three months, so this is a drop of 4 per month. Starting from £3,000, this requires 1 month to become unprofitable, so in month 5 of the year – assuming Q2 starts in month 4.

** Step 2:** Similarly, for the projected profit, between Q3 and Q4 projected profit changes by -18. This is a drop of 6 per month. The company becomes unprofitable in month 2 of Q3, or the 8

^{th}month of the year.

** Step 3:** The difference in these is 8-4 = 4 months

**Video Solutions**

**Question 8**

Calculate actual costs as a proportion of revenue for 2018. Give your answer to 1dp.

A: 85.8%

B: 87.0%

C: 88.2%

**D: 89.4%**

**Written Solutions**

** Step 1:** Add up total revenue.

29+33+24+18=\pounds104,000

** Step 2:** Calculate and add up total costs. Calculate cost by subtracting profit from revenue.

(29-24)+(33-3)+(24--9)+(13--12)=\pounds93,000

** Step 3:** Next calculate costs as a proportion of revenue.

\dfrac{93,000}{104,000}\times100=\bold{89.4\%}

**Video Solutions**

**Question 9**

Business tax must be paid yearly at the stated rate for every £1 of rateable value. Id payments are spread out evenly over the four quarters, what is the overall percentage change in profit of Super Fry from Q1 to Q3?

**A: 28.2% decrease**

B: 31.4% decrease

C: 24.1% decrease

D: 33.7% decrease

**Written Solutions**

** Step 1:** First calculate the business tax owed and divide this by four. Multiply by 0.486 for a tax rate of 48.6p per £1.

24,000 \times 0.486 = \pounds11,664

11,664 \div 4 = \pounds2,916\text{ per quarter}

** Step 2:** Calculate quarterly costs. These are equal for both quarters.

12,000+2,916=\pounds14,916

** Step 3:** Find the profit for each quarter, then calculate the percentage change.

\text{Q1: }22,000-14,916=\pounds7,084

\text{Q3: }20,000-14,916=\pounds5,084

\dfrac{5,084-7,084}{7,084}\times100=\bold{-28.2\%\text{ or a 28.2\% decrease}}

**Video Solutions**

**Question 10**

Without considering business tax, calculate the total profit for each business in 2018. Express this as a ratio in its lowest terms.

A: 1.37: 1: 1.25

**B: 1.39: 1: 1.29**

C: 1.41: 1: 1.33

D: 1.45: 1: 1.37

**Written Solutions**

** Step 1:** First calculate the total revenue for each company.

\begin{aligned}\text{Apples2Apples: }&17+23+25+14=\pounds79,000 \\ \text{Good Fortune: }&13+16+27+35=\pounds91,000 \\ \text{Super Fry: }&22+19+20+27=\pounds88,000\end{aligned}

** Step 2:** Subtract the costs from revenue to get profit. We need to multiply quarterly costs by 4 first to cover the whole year.

\begin{aligned}\text{Apples2Apples: }&79 - (9\times 4) = \pounds43,000 \\ \text{Good Fortune: }&91 - (15 \times 4) = \pounds31,000 \\ \text{Super Fry: }&88 - (12 \times 4) = \pounds40,000\end{aligned}

** Step 3:** Write these as a ratio and simplify.

\begin{gathered}\text{Apples2Apples}\colon\text{Good Fortune}\colon\text{Super Fry} \\ \pounds43,000\colon\pounds31,000\colon\pounds40,000 \\ \dfrac{43,000}{31,000}\colon1\colon\dfrac{40,000}{31,000} \\ \bold{1.39\colon1\colon1.29}\end{gathered}

**Video Solutions**

**Question 11**

For the quarter with the highest overall revenue, express total profits over all three businesses (not including business tax costs) as a proportion of total revenue.

A: 51.6%

B: 52.0%

**C: 52.6%**

D: 52.8%

**Written Solutions**

** Step 1:** First we need to decide which quarter has the highest revenue. By inspection it seems to be either Q3 or Q4. Add up the revenues for these to compare between the two.

\text{Q3: }25+27+20=72

\text{Q4: }14+35+27=76

Q4 is higher so we will calculate for Q4.

** Step 2:** Add up total costs for Q4, this is the sum of quarterly costs.

9+15+12=36

Now calculate the total profit by subtracting this from revenue.

76-36=40

** Step 3:** Express this as a proportion of the total.

\dfrac{40}{76}\times100=\bold{52.6\%}

**Video Solutions**

**Question 12**

Including business tax charged on every £1 of rateable value, which business was the most profitable in 2018?

**A: Apples2Apples**

B: Good Fortune

C: Super Fry

D: Cannot tell

**Written Solutions**

** Step 1:** Calculate total revenue for each business.

\begin{aligned}\text{Apples2Apples: }&17+23+25+14=\pounds79,000 \\ \text{Good Fortune: }&13+16+27+35=\pounds91,000 \\ \text{Super Fry: }&22+19+20+27=\pounds88,000\end{aligned}

** Step 2:** Next calculate the business tax for each company. This is charged at 48.6p per £1, or a multiplier of 0.486.

\begin{aligned}\text{Apples2Apples: }&0.486\times10000=\pounds4,860 \\ \text{Good Fortune: }&0.486\times32000=\pounds15,552 \\ \text{Super Fry: }&0.486\times24,000=\pounds11,664\end{aligned}

** Step 3:** Now subtract quarterly costs (multiplied by 4 to cover the whole year), and subtract the business tax.

\begin{aligned}\text{Apples2Apples: }&79,000 - (9,000\times 4)-4,860= \bold{\pounds38,140} \\ \text{Good Fortune: }&91,000 - (15,000 \times 4)-15,552=\pounds15,448 \\ \text{Super Fry: }&88,000 - (12,000 \times 4)-11,664= \pounds28,336\end{aligned}

So Apples2Apples was the most profitable in 2018.

**Video Solutions**

**Question 13**

A company needs to send nine employees in three groups from Harrogate to each of the following three locations: Clifton Moor, Malton and Scarborough. What is the cheapest total cost to the company for return tickets?

A: £81.45

B: £84.25

C: £86.54

**D: £87.48**

**Written Solutions**

** Step 1:** Calculate the cost for the three that go to Clifton Moor. The cheapest route uses the number 72B, since there are only 2 stops to Clifton Moor. 2 stops costs £1.20 per stop, and for a return we multiply this price by 1.9 to get the price per person.

2 \times \pounds1.20 \times 1.9 = \pounds4.56

Now multiply by three to get the group price.

\pounds4.56 \times 3 = \pounds13.68

** Step 2:** The only route to Malton is on the 72A and takes 4 stops. At £1.20 per stop, for three people, with a multiplier of 1.9 for a return we get the following price

\pounds1.20 \times 4 \times 3 \times 1.9 = \pounds27.36

** Step 3:** For the Scarborough journey, the full ticket price of £8.60 can be bought. For three people, and a multiplier of 1.8 for a return we now calculate the price

\pounds8.60 \times 3 \times 1.8 = \pounds46.44

** Step 4:** Finally add up all these costs to find the total cost

13.68+27.36+46.44=\bold{\pounds87.48}

**Video Solutions**

**Question 14**

Starting at 9am, a shopper first spends 1:15 in Harrogate before getting the next bus to Clifton Moor. They then spend 2.25 hours there and get the next bus to Monks Cross. After 165 minutes they get the next bus to Staxton where they end their journey. At what time do they finish their journey?

A: 16:40

B: 16:52

**C: 17:40**

D: 17:52

**Written Solutions**

** Step 1:** After 1:15 in Harrogate the time is 10:15. The next bus to Clifton Moor is on the 72, from 10:54 to 11:40.

** Step 2:** 2.25 hours on from 11:40 is 13:55, and the next bus from Clifton Moor to Monks Cross is on the 72B, from 13:57 – 14:22.

** Step 3:** 165 minutes after 14:22 is 17:07. The next bus to Staxton is the 17:12 which arrives in Staxton at

**17:40.**

**Video Solutions**

**Question 15**

An audit of the bus service occurs. What is the average duration of all journeys in one day from Harrogate to Staxton?

A: 2:03

**B: 2:05**

C: 2:09

D: 2:11

**Written Solutions**

** Step 1:** For the first journey of the day on the 72, the journey time is 10:25 – 08:45 = 1:40. For the 72A, the journey time is 11:03 – 08:57 = 2:06.

** Step 2:** For the journeys that recur during the middle of the day, the duration for the 72 is 1:46. For the 72A the duration is 2:21. These occur 10 times during the day.

** Step 3:** For the final journey of each day, the duration on the 72 is 22:00 – 19:54 = 2:06. For the 72A the duration is 22:25-19:31=2:54

** Step 4:** Now we need to find the average. Remember there are 10 repeats of the journeys in the middle of the day. The total duration is

1:40+2:06+10(1:46+2:21)+2:06+2:54=49:56

Now divide this by the total number of journeys, which is 24.

49:56 \div 24 = \bold{02:05}

**Video Solutions**

**Question 16**

On the final bus on each route, what is the speed of the slowest journey from Harrogate to Earswick?

A: 18.8mph

B: 19.0mph

**C: 19.2mph**

D: 19.4mph

**Written Solutions**

** Step 1:** Check the final three journeys to find the one that takes the longest. This is the 72B, that takes 01:15 or 75 minutes.

** Step 2:** Now calculate the distance from Harrogate to Earswick.

4+8+9+3=24\text{ miles}

** Step 3:** Now calculate the average speed over this distance

\text{Speed}=\dfrac{\text{Distance}}{\text{Time}}=\dfrac{24}{75}\times60=\bold{19.2\text{mph}}

**Video Solutions**

**Question 17**

When leasing an initial payment must be made. In this case it accounts for 57% of the Y1 payment. If this value is removed from the total lease payments, what is the percentage increase in total lease payments over 5 years? Give your answer to the nearest whole number.

A: 504%

**B: 517%**

C: 530%

D: 543%

**Written Solutions**

** Step 1:** First calculate 57% of the Y1 payment. This is a multiplier of 0.57

0.57 \times 1800=\pounds1,026

** Step 2:** Subtract this from the Y1 and Y5 payments.

\text{Y1: }1800-1026=\pounds774

\text{Y5: }5800-1026 = \pounds4,774

** Step 3:** Now calculate the percentage increase over this period.

\dfrac{4774-774}{774}\times100=\bold{517\%}

**Video Solutions**

**Question 18**

What is the ratio of upkeep costs in Y1 compared to Y2? Give your answer in its simplest form.

**A: 1.17: 1**

B: 1.19: 1

C: 1.21: 1

D: 1.23: 1

**Written Solutions**

** Step 1:** First calculate the total Year 1 upkeep costs. We need 10% of the value of the car which is a multiplier of 0.1

(0.1\times25,000)+50+145+120=\pounds2,815

** Step 2:** Now do the same calculation for Year 2

(0.1 \times 21,000) + 50+145+120=\pounds2,415

** Step 3:** Write the ratio and simplify.

\begin{gathered}\text{Year 1}\colon\text{Year 2} \\ \pounds2,815\colon\pounds2,415 \\ \dfrac{\pounds2,815}{\pounds2,415}\colon1 \\ \bold{1.17\colon1} \end{gathered}

**Video Solutions**

**Question 19**

If the trend in total lease payments from Y4 to Y5 continues to increase, after how many more years will the lease payment be more than the value of a new €20,000 car, if the exchange rate is £1 to €1.25?

A: 2

B: 3

**C: 4**

D: 5

**Written Solutions**

** Step 1:** Find the percentage increase in lease payments from Y4 to Y5.

\dfrac{5.8-4.3}{4.3}\times100=34.9\% (34.8837…)

** Step 2:** Convert the price of the car into pounds.

20,000 \div 1.25 = \pounds16,000

** Step 3:** Now write an expression for the increase in total lease payments. We do not know the year so call this year n.

5,800\times1.349^n

We want this to be greater than the cost of the car

5,800\times1.349^n > 16,000

Substitute values of n to find the lowest number that makes the right-hand side greater than 16,000 – this is after **4 years.**

**Video Solutions**

**Question 20**

What is the total cost of upkeep for a car as the percentage of its value in Year 5?

A: 17.03%

B: 17.23%

C: 17.34%

**D: 17.58%**

**Written Solutions**

** Step 1:** Calculate new road tax. This is a multiplier of 1.2 for each year after year 3, so the calculation is

50\times1.2^2=\pounds72

** Step 2:** Breakdown cover is £145 plus two lots of £25, so

**£195**. For the minor repairs we make the same calculation as road tax, but with a multiplier of 1.15.

120\times1.15^2=\pounds158.70

** Step 3:** Now calculate 15% of the value of the car. 15% is a multiplier of 0.15.

0.15\times 16,500=\pounds2,475

** Step 4:** Finally add up these costs and find it as a percentage of the total value of £16,500.

72+195+158.70+2,475=\pounds2,900.70

\dfrac{2900.70}{16500}\times100=\bold{17.58\%}