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Worked Solutions – Test 6 2018-11-01T17:54:56+00:00

Worked Solutions – Test 6

Question 1 From 2017 to 2018, what was the percentage decrease in value of combined Christmas and January sales? Give your answer to 1 decimal place.

A: 19.3%

B: 19.5%

C: 19.7%

D: 19.8%

Written Solutions

Step 1: Calculate the combined percentage share of sales for 2017, then calculate the value this represents.

$21\% + 38\% = 59\%$

$59\%\text{ of }120 = 0.59 \times 120 = \pounds70.8\text{m}$

Step 2: Make the same calculation for 2018

$11\% + 27\% = 38\%$

$38\%\text{ of }150 = 0.38 \times 150 = \pounds57\text{m}$

Step 3: Calculate the percentage decrease from 2017 to 2018

$\dfrac{57-70.8}{70.8} \times 100 = -19.5\%\text{ (1dp)}$

Question 2 Given that the ratio of overall sales in 2017, 2018 and 2019 is 4: 5: 7, and the ratio of the percentage share of sales for Back to School is 6: 4: 5. Calculate the value of the back to school season for 2019.

A: £17.5m

B: £18m

C: £21m

D: £22m

Written Solutions

Step 1: We need to figure out what to multiply the total sales ratio by to get the actual figures. First compare the share for 2017 (4) to the actual value (120m).

$\dfrac{120}{4}=30$

The actual values are 30 times bigger than the ratio (in millions). Now multiply the ratio by 30 to discover the actual values for each year.

$4: 5: 7 \equiv 120: 150: 210$

So, the total sales for 2019 are £210m

Step 2: Use the same process to calculate the percentage share of sales represented by back to school in 2019.

$\dfrac{12\%}{6} = 2$

So multiply the ratio by two to get the percentage shares for each year

$6: 4: 5 \equiv 12\%: 8\%: 10\%$

Step 3: We know back to school is 10% of the total sales for 2019 so we need to find:

$10\%\text{ of }210\text{m} = 0.10 \times 210 = \bold{\pounds21\text{m}}$

Question 3 Which period had the largest change in value from 2017 to 2018?

A: January Sales

B: Easter

C: Summer

D: Christmas

Written Solutions

Step 1: Find the value of each shopping period in 2017

\begin{aligned}\text{January Sales: }&0.21 \times 120 = 25.2 \\ \text{Easter: }&0.14 \times 120 = 16.8 \\ \text{Summer: }&0.15 \times 120 = 18 \\ \text{Christmas: }& 0.38 \times 120 = 45.6\end{aligned}

Step 2: Find the value of each shopping period in 2018

\begin{aligned}\text{January Sales: }&0.11 \times 150 = 16.5 \\ \text{Easter: }&0.26 \times 150 = 39 \\ \text{Summer: }&0.28 \times 150 = 42 \\ \text{Christmas: }& 0.27 \times 150 = 40.5\end{aligned}

Step 3: Now calculate the difference between the values for 2017 and 2018.

\begin{aligned}\text{January Sales: }&25.2 - 16.5 = \pounds8.7\text{m} \\ \text{Easter: }&39 - 16.8 = \pounds22.2\text{m} \\ \text{Summer: }&42 - 18 = \bold{\pounds24\text{m}} \\ \text{Christmas: }&45.6 - 40.5 = \pounds5.1\text{m} \end{aligned}

Question 4 Use the prediction to calculate the combined value of January Sales & Back to School in 2019 if the total sales from 2018 to 2019 increase by 22%.

A: £11.22m

B: £13.45m

C: £15.62m

D: £17.90m

Written Solutions

Step 1: Calculate the total value of all sales in 2019. A 22% increase is a multiplier of 1.22.

$1.22 \times 150 = \pounds183\text{m}$

Step 2: Calculate the sales of Easter, Summer and Christmas respectively.

\begin{aligned}\text{Easter: }&(0.26\times 150) + 12 = \pounds61\text{m} \\ \text{Summer: }&0.28 \times 183 = \pounds51.24\text{m} \\ \text{Christmas: }&0.38 \times 183 = \pounds69.54\text{m}\end{aligned}

Step 3: Subtract these from the total value to find the combined value of the two desired sales.

$183 - (61 + 51.24 + 69.54) = \pounds11.22\text{m}$

Question 5 On a G&O cruise, what is the ratio of time spent on the boat to time spent in cities?

A: 3.5: 1

B: 3.0: 1

C: 3.4: 1

D: 3.2: 1

Written Solutions

Step 1: Add up the time spent on the boat. This is the overnight periods – for example the first section from 19:00 to 11:00 is on the boat. Make sure to convert times such as 03:45 into a decimal – 3.75.

$11+5+1.7515+1.5+15.5+2+13.75+3.5+24+9.5 = 102.5\text{ hours}$

Step 2: Next add up the time spent in cities, this is the time during the day. Again make sure to convert times into decimals.

$11.25+7.5+6.5+6.75 = 32\text{ hours}$

Step 3: Finally write the ratio of boat:cities and cancel it down.

$\begin{gathered}102.5:32 \\ \dfrac{102.5}{32}:\dfrac{102.5}{102.5} \\ 3.2:1\end{gathered}$

Question 6 Which provider offers the best average price per day for someone who buys a standard ticket and an extension?

A: Sun Cruiser

B: G&O

C: Nordic

D: River Line

Written Solutions

Step 1: For each provider, add the standard price to the extension price

\begin{aligned}\text{Sun Cruiser: }&2990+1700 = 4690 \\ \text{G\&O: }&2580+800 = 3380 \\ \text{Nordic: }&2790+650 = 3440 \\ \text{River Line: }&3140+570=3710\end{aligned}

Step 2: Also calculate the number of days that each cruise lasts for when taking into account the extension.

\begin{aligned}\text{Sun Cruiser: }&3+9=12\\ \text{G\&O: }&2+9=11 \\ \text{Nordic: }&2+9=11 \\ \text{River Line: }&1+9=10 \end{aligned}

Step 3: Divide the cost by the number of days for each provider to calculate the average and select the best value.

\begin{aligned}\text{Sun Cruiser: }&\dfrac{4690}{12}\approx\pounds391 \\ \text{G\&O: }&\dfrac{3380}{11}\approx\bold{\pounds307} \\ \text{Nordic: }&\dfrac{3440}{11}\approx\pounds313 \\ \text{River Line: }&\dfrac{3710}{10}\approx\pounds371\end{aligned}

Question 7 Calculate the average speed for the river line cruise over the full duration of the trip.

A: 10.2mph

B: 10.6mph

C: 11.0mph

D: 11.4mph

Written Solutions

Step 1: Find the total distance travelled.

$80+400+55+300+510+180=1525\text{ miles}$

Step 2: Find the total time spent. Add on 5 full days and the additional time on day 1 and day 9.

$12.5+(5 \times 24)+11=143.5\text{ hours}$

Step 3: Finally use the formula for speed to calculate the average speed.

$\text{Speed} = \dfrac{\text{Distance}}{\text{Time}} = \dfrac{1525}{143.5} \approx \bold{10.6\text{mph}}$

Question 8 On a Sun Cruiser deluxe cruise, time spent in Dubrovnik and Corfu costs an average of £12 per hour in tour fees. If all time spent at these locations is spent touring, what is the percentage increase in overall cost due to this addition?

A: 6.12%

B: 6.34%

C: 6.57%

D: 6.63%

Written Solutions

Step 1: Calculate the time spent in Dubrovnik and Corfu.

$19:30-05:00 = 14:30 \equiv 14.5\text{ hours}$

$21:30-10:00 = 11:30 \equiv 11.5\text{ hours}$

Step 2: Next multiply this time spent by the cost of an hours touring and add this to the deluxe price.

$4750+(12\times26)=\pounds5062$

Step 3: Now calculate the percentage increase

$\dfrac{5062 - 4750}{4750} \times 100 = \bold{6.57\%}$

Question 9 What is the percentage decrease in profit from April to July?

A: 96%

B: 95%

C: 94%

D: 93%

Written Solutions

Step 1: Calculate profit in April by taking the costs away from revenue.

$450 - 200 = \pounds250\text{m}$

Step 2: Calculate profit in July by taking the costs away from revenue.

$240 - 230 = \pounds10\text{m}$

Step 3: Calculate the percentage decrease.

$\dfrac{250-10}{250}=96\%$

Question 10 1 store closed reduces costs for that month by £1.95m. If stores had not closed over the period in which month would costs have overtaken revenue?

A: March

B: April

C: May

D: June

Written Solutions

Step 1: Calculate the new costs for each month. Multiply the number of stores closed by £1.95m.

\begin{aligned}\text{March: }&220+(1.95 \times120)=\pounds454\text{m} \\ \text{April: }&200+(1.95 \times85)=\pounds365.8\text{m} \\ \text{May: }&150+(1.95 \times90)=\pounds325.5\text{m} \\ \text{June: }&200+(1.95 \times115)=\pounds424.3\text{m}\end{aligned}

Step 2: Now take the costs from the revenue to find the month that costs are greater than revenue.

\begin{aligned}\text{March: }&520-454=\pounds66\text{m} \\ \text{April: }&450-365.8=\pounds84.3\text{m} \\ \text{May: }&320-325.5=\bold{\pounds-5.5\text{m}} \\ \text{June: }&290-424.3=\pounds-134.3\text{m}\end{aligned}

Question 11 Provided that the percentage changes in revenue & costs from June to July continue, make a prediction for the profit in August.

A: £57.45m loss

B: £18.24m profit

C: £65.78m loss

D: £3.41m profit

Written Solutions

Step 1: Calculate the percentage change in revenue.

$\dfrac{240-290}{290}\times100\approx-17.2\%$

Step 2: Calculate the percentage change in costs.

$\dfrac{230-200}{200}\times100=15\%$

Step 3: Apply the percentage increase and decrease to the July figures to calculate revenue and cost in August. From there find the difference to calculate profit.

-17.2% is equivalent to finding 82.8% or a multiplier of 0.828

$0.828\times240=\pounds198.72\text{m}$

15% increase is equivalent to finding 115% or a multiplier of 1.15

$1.15\times230=\pounds264.5\text{m}$

Profit is calculated by taking costs away from revenue so we find:

$198.72-264.5=\bold{-\pounds65.78\text{m}}\text{ - in other words a loss of }\pounds65.78\text{m}$

Question 12 What was the difference in profit as a proportion of revenue between April and May?

A: 1.9%

B: 2.1%

C: 2.3%

D: 2.5%

Written Solutions

Step 1: Calculate profit for April and express this as a proportion of revenue

$\text{Profit }=450-200=\pounds250\text{m}$

$\dfrac{250}{450}\times100\approx55.6\%$

Step 2: Calculate profit for May and express this as a proportion of revenue

$\text{Profit }=320-150=\pounds170\text{m}$

$\dfrac{170}{320}\times100\approx53.1\%$

Step 3: Find the difference

$55.6\%-53.1\%=\bold{2.5\%}$

Question 13 What was the percentage increase in total take home pay for Ben & Celica from June to July?

A: 6.9%

B: 7.2%

C: 7.5%

D: 7.8%

Written Solutions

Step 1: Find the sum of take home pay in June from Ben & Celica

$1240+1360=\pounds2600$

Step 2: Find the sum of take home pay in July from Ben & Celica

$1320+1460=\pounds2780$

Step 3: Now find the percentage increase

$\dfrac{2780-2600}{2600}\times100=\bold{6.9\%}$

Question 14 The ratio of pay for Anna, Ben and Celica in September was 9: 6: 5. If Celica earned €2250 in September, what was the total earnings for all three employees in pounds?

A: £7080

B: £7200

C: £7500

D: £7650

Written Solutions

Step 1: First convert Celica’s wage into pounds. To do this, divide her wage by the exchange rate.

$\dfrac{2250}{1.25}=\pounds1800$

Step 2: Next use the ratio to find out the wages of the three employees. We see that in the ratio, Celica’s share is 5. Divide her wage by the share of the ratio to work out how many times larger the wages are than the ratio.

$1800 \div 5 = 360$

Now multiply the ratio by 360 to get the actual value of wages.

$9: 6: 5 \equiv 3240: 2160: 1800$

Step 3: Now that we know the actual wages, add these up to find the total.

$3240+2160+1800=\bold{\pounds7200}$

Question 15 Calculate Dan’s take-home pay in June in pounds, given that he first pays tax of 20% on his full base income.

A: £985

B: £1005

C: £1045

D: £1070

Written Solutions

Step 1: First reduce Dan’s basic income for June by 20% to account for the tax he pays. We are finding 80% or a multiplier of 0.80

$0.8\times1125=900$

Step 2: We need to know what the value of one tip is. There are $19+37+24=80$ tips in total and we know the total value of this, so divide to find the value of one tip.

$1500 \div 80 = 18.75$

Now we know the value of one tip, we can calculate the value of 19 tips in June.

$19 \times 18.75 = 356.25$

Step 3: Now add the reduce base income and tips to find the total income for June.

$900+356.25=1256.25\text{ euros}$

Finally, we need to convert into pounds, so divide by the exchange rate.

$1256.25 \div 1.25 = \bold{\pounds1005}$

Question 16 What is the difference in Dan’s and Anna’s total pay over the full 3 months in pounds?

A: £580

B: £600

C: £620

D: £640

Written Solutions

Step 1: Add up Anna’s pay to calculate her total

$1400+1600+1580=\pounds4580$

Step 2: Find Dan’s total pay and convert into euros. We know over the full three months he earned 1500 euros in tips.

$1125+1175+1150+1500=4950\text{ euros}$

$4950 \div 1.25 = \pounds3960$

Step 3: Find the difference now that we have both wages in pounds

$4580-3960=\bold{\pounds620}$

Question 17 A buyer switches from considering a %5 mortgage on a £180,000 house to a 15% mortgage. What is the percentage change in the total initial cost to the nearest whole number?

A: 163%

B: 165%

C: 167%

D: 169%

Written Solutions

Step 1: Calculate the initial cost for the 5% mortgage. We need to calculate 5% (or a multiplier of 0.05) of 180,000 and add on the other initial costs.

$(0.05\times180,000)+1,800=\pounds10,800$

Step 2: Make the same calculation for a 15% mortgage.

$(0.15\times180,000)+2,100=\pounds29,100$

Step 3: Calculate the percentage change.

$\dfrac{29100-10800}{10800}\times100\approx\bold{169\%}$

Question 18 What is the difference between the average monthly payment for a £240,000 house compared to a £300,000 house?

A: £226

B: £231

C: £236

D: £241

Written Solutions

Step 1: Add up the monthly payments for a £240,000 house and calculate the average.

$1259+1147+1071+996+883+802+866+757+679=\pounds8,460$

$8460 \div 9 = \pounds940$

Step 2: Make the same calculation for a £300,000 house.

$1574+1430+1345+1245+1105+1000+1089+946+850=\pounds10,584$

$10584 \div 9 = \pounds1,176$

Step 3: Find the difference

$1176-940=\pounds236$

Question 19 Someone takes out a 10% 30-year mortgage for a £240,000 house. Express the total amount paid over the full 30 years as a proportion of the cost of the house to the nearest whole number.

A: 140%

B: 144%

C: 148%

D: 152%

Written Solutions

Step 1: Calculate initial payments. This is 10% of the cost of the house plus other costs.

$(0.1\times240,000)+2,650=\pounds26,650$

Step 2: Now calculate the total spent on repayments over 30 years. From the table the monthly repayment is £883.

$883 \times 30 \times 12 = \pounds317,880$

Step 3: We now calculate the total amount paid. After that we can calculate this as a proportion of the original cost of the house.

$317,880+26,650=\pounds344,530$

$\dfrac{344530}{240000}\times100\approx\bold{144\%}$

Question 20 A first-time buyer earns £25,000 per year and saves 18% of this each year. How long will it take to save up the deposit and other initial costs and pay off a 15% loan on a £240,000 house paid over 30 years.

A: 36 years

B: 37 years

C: 38 years

D: 39 years

Written Solutions

Step 1: First we need to know how much a deposit costs. Find 15% of 240,000, or a multiplier of 0.15. Then add on the other initial costs.

$0.15\times240,000=\pounds36,000$

$36,000+3,000=39,000$

Step 2: Now calculate how much the buyer can save per year and figure out how many years it will take to save up this deposit. 18% is a multiplier of 0.18

$0.18 \times25,000 = \pounds4,500$

$39000 \div 4500 = 8.67\text{ years or }9\text{ full years}$

Step 3: Now add this to the duration of the loan to calculate how long it takes in total.

$9+30=\bold{39\text{ years}}$