Question 1

In 2012, 3.5% of the Stilton that was sold had to later be fully refunded due to faulty packaging. What was the percentage increase in the profit from sales of Stilton from 2012 to 2013, considering that in both years Stilton made a profit of £0.85 per pound sold?

A: 9.46%

B: 13.05%

C: 12.87%

D: 6.90%

Written Solutions

Step 1: Remove the 3.5% of Stilton, on which no profit was made, from the total amount to obtain the profitable amount in kg.

3.5\% corresponds to a multiplier of 0.035. So, the profitable amount of Stilton sold is

11,000 - (0.035 \times 11,000) = 10,615kg

Step 2: Firstly, multiply by 2.2 to convert the kg to pounds, and then multiply by 0.85 to determine, in £, the profit from sale of Stilton for each year.

2012: (10,615 \times 2.2) \times 0.85 = 23,353 \times 0.85 = \pounds19,850.05 of profit;

2013: (12,000 \times 2.2) \times 0.85 = 26,400 \times 0.85 = \pounds 22,440 of profit.

Step 3: Finally, use these values to determine the percentage increase as such.

\dfrac{22,440 - 19,850.05}{19,850.05} \times 100 = 13.05 \%

Question 2

8 litres of milk are required to make one kilogram of Cheddar, whilst 2.5 litres of milk are required to make 1 pound of Mozzarella. Determine the ratio of the milk used to make Cheddar compared to the milk used to make Mozzarella over the span of 2011 and 2012.

A: 1.30: 1

B: 1.52: 1

C: 1: 1.98

D: 1: 1.64

Written Solutions

Step 1: Determine the total sales for both Cheddar and Mozzarella over 2011 and 2012.

Cheddar: 24,000 + 23,000 = 47,000\text{ kg}

Mozzarella: 23,000 + 22,000 = 45,000\text{ kg}

Step 2: Calculate the litres required to make these amounts of each cheese. The information in the question gives us “litres per pound” for Mozzarella, so we should firstly convert our amount of Mozzarella into pounds.

45,000 \times 2.2 = 99,000 pounds of Mozzarella. Then, we get the following.

\text{Milk used for Cheddar } = 47,000 \times 8 = 376,000 \text{ litres}

\text{Milk used for Mozzarella } = 99,000 \times 2.5 = 247,500 \text{ litres}

Step 3: Now, form the ratio and simplify it down by dividing by the smallest amount. See:

\begin{aligned}\text{Cheddar : Mozzarella } &= 376,000 : 247,500 \\ &= \dfrac{376,000}{247,500} : \dfrac{247,500}{247,500} \\ &= 1.52 : 1\end{aligned}

Question 3

The percentage increase/decrease for sales of Vegan Cheese and Cheddar respectively, from 2012-2013, are both expected to continue for 2 years after 2013. According to this prediction, what will be the difference, to the nearest pound, between the weights of Cheddar and Vegan Cheese sold in 2015?

A: 2374 pounds

B: 2458 pounds

C: 2111 pounds

D: 3016 pounds

Written Solutions

Step 1: Calculate the percentage changes for the two cheeses.

Vegan: \dfrac{11,000 - 9,000}{9,000} \times 100 = 22.2\% \text{ increase}. This corresponds to the multiplier 1.222.

Cheddar: \dfrac{23,000 - 21,000}{23,000} \times 100 = 8.70\% \text{ decrease}. This corresponds to the multiplier 0.913.

Step 2: Work out the sales of each cheese after a further 2 years at this same rate of percentage cheese. Note, we take each percentage multiplier to the power of 2 to account for 2 years of change.

Vegan: 11,000 \times 1.222^2 = 16426 \text{ kg}

Cheddar: 21,000 \times 0.913^2 =17505 \text{ kg}

Step 3: Determine the difference in sales and multiply by 2.2 to convert to pounds.

(17505- 16426) \times 2.2 = 2374 \text{ pounds, to the nearest pound.}

Question 4

The exchange rate in 2011 and 2012 was £1 = $1.40. By 2013, the value of the pound on the dollar had decreased by 5%. The profit on sales of Mozzarella remained consistent across all 3 years at £1.52 per kg. What is the ratio of profits on sales of Mozzarella in dollars, comparing 2011 to 2012 to 2013?

A: 1.07: 1: 1.09

B: 1.22: 1: 1.28

C: 1.13: 1: 1.08

D: 1.05: 1: 1.12

Written Solutions

Step 1: Firstly, work out what the exchange rate between pounds and dollars would be in 2013.

A 5% decrease means the multiplier is 0.95, so 1.40 \times 0.95 = 1.33, thus the exchange rate for 2013 is: £1 = $1.33.

Step 2: Calculate the profits for each year in pounds, and then convert to dollars.

\text{2011 profit } = 23,000 \times 1.52 = \pounds 34,960 \text{ which is equal to } 34,960 \times 1.40 = \$ 48,944

\text{2012 profit } = 22,000 \times 1.52 = \pounds 33,440 \text{ which is equal to } 33,440 \times 1.40 = \$ 46,816

\text{2013 profit } = 26,000 \times 1.52 = \pounds 39,520 \text{ which is equal to } 39,520 \times 1.33 = \$ 52561.60

Step 3: Form the ratio and simplify by dividing through by the smallest amount.

\begin{aligned} \text{2011 : 2012 : 2013 } &= 48,944 : 46,816 : 52,561.60 \\ &= \dfrac{48,944}{46,816} : \dfrac{46,816}{46,816} : \dfrac{52,561.60}{46,816} \\ &= 1.05 : 1 : 1.12 \end{aligned}

Question 5

What is the difference between the average monthly costs of a contract with the two providers whose upload speeds are at least 25% as high as their download speeds? You may assume no data limits are exceeded.

A: £1.02

B: £0.82

C: £0.76

D: £0.12

Written Solutions

Step 1: Work out which two providers have adequate upload speeds.

\begin{aligned}\text{Webby: }& 25\% \text{ of } 15 = 0.25 \times 15 = 3.75 > 3 \text{ mbps} \\ \text{NetUK: }& 25\% \text{ of } 15 = 0.25 \times 15 = 3.75 < 5 \text{ mbps } \checkmark \\ \text{Surf4Less: }& 25\% \text{ of } 35 = 0.25 \times 35 = 8.75 < 12 \text{ mbps } \checkmark \\ \text{WyFy: }& 25\% \text{ of } 32 = 0.25 \times 32 = 8 > 7 \text{ mbps} \end{aligned}

Step 2: Work out the total cost of a contract with NetUK and Surf4Less.

\text{NetUK, total cost } = 56 + (12 \times 18.99) = \pounds 283.88

\text{Surf4Less, total cost } = 22.95 + (24 \times 23.52) = \pounds 587.43

Step 3: Calculate the average monthly costs, and subtract one from the other to find the difference.

\text{NetUK, average monthly cost } = \dfrac{283.88}{12} = \pounds23.66

\text{Surf4Less, average monthly cost } = \dfrac{587.43}{24} = \pounds24.48

\text{Difference } = 24.48 - 23.66 = \pounds 0.82

Question 6

The cost of a monthly payment listed, the 2017 cost, for a Surf4Less contract is a 20% increase on the monthly payment for the same contract in 2016. Given that the installation costs have remained unchanged for at least the last 2 years, what is the percentage change in the total cost of this Surf4Less contract?

A: 19.1%

B: 18.3%

C: 18.1%

D: 17.4%

Written Solutions

Step 1: Work out the 2016 cost of installing Surf4Less internet.

A 20% increase on the previous cost means that the £23.52 must be the result of multiplying the previous value by 1.20 (the 20% multiplier). So, to find the previous year’s cost, we must then divide by 1.20. See:

23.52 \div 1.20 = \pounds 19.60 \text{ per month}

Step 2: Calculate the total cost of the contract in this year and the previous year.

\text{2017 cost } = 22.95 + (24 \times 23.52) = \pounds 587.43

\text{2016 cost } = 22.95 + (24 \times 19.60) = \pounds 493.35

Step 3: Use these values to determine the percentage increase from 2016 to 2017.

\text{Percentage increase } = \dfrac{587.43 - 493.35}{493.35} \times 100 = 19.1\%

Question 7

I will use 370gb of data every month. Work out the ratio of the average monthly cost of a Webby contract for me compared to the average monthly cost of a WyFy contract for me.

A: 1: 1.02

B: 1: 1.15

C: 1.13: 1

D: 1.02: 1

Written Solutions

Step 1: Work out the total cost of the contracts of both Webby and WyFy.

Firstly, we must determine the effects of using 370gb of data. This is 20gb over the 350gb allowance for Webby, which means the monthly cost (for every month) will increase by £5 to £25.50. Now, we must see, for WyFy, if we will use less than 8,000gb by the end of the contract.

370 \times 18 = 6,660 < 8,800 \text{ gb}, which tells us that WyFy will refund the cost of the installation. So, we get:

\text{Webby: total cost } = 25 + (12 \times 25.50) = \pounds 331

\text{WyFy: total cost } = 18 \times 27 = \pounds486

Step 2: Find the average monthly cost for both providers.

\text{Webby: average monthly cost } = 331 \div 12 = \pounds 27.58

\text{WyFy: average monthly cost } = 486 \div 18 = \pounds 27

(You may notice because the installation cost was refunded by WyFy, we could’ve alternatively determined straight away that the average monthly cost was just £27.)

Step 3: Form the ratio and simplify it by dividing through by the smallest value.

\begin{aligned}\text{Webby : WyFy } &= 27.58 : 27 \\ &= \dfrac{27.58}{27} : \dfrac{27}{27} \\ &= 1.02 : 1\end{aligned}

Question 8

Next year the NetUK charge added for exceeding the 500gb data limit would become an extra £6 for every 50gb of data over the limit (every month), whilst the regular monthly payment would increase to £20.99 per month. By what percentage does this decrease the cost of the contract for someone who will use 600gb of data per month?

A: 13.7%

B: 9.5%

C: 12.1%

D: 6.4%

Written Solutions

Step 1: Considering 600gb of data will be used every month, calculate the new monthly payment (including the surcharges) for both the current NetUK contract and next year’s proposed NetUK contract.

600gb is 100gb over the limit.

100 = 20 \times 5 \text{ so, current monthly payment } = 18.99 + (5 \times 4) = \pounds 38.99

100 = 50 \times 2 \text{ so, proposed monthly payment } = 20.99 + (2 \times 6) = \pounds 32.99

Step 2: Work out the total cost of the current contract and the proposed contract.

\text{Current contract cost } = 56 + (12 \times 38.99) = \pounds 523.88

\text{Proposed contract cost } = 56 + (12 \times 32.99) = \pounds 451.88

Step 3: Calculate the percentage decrease.

\dfrac{523.88 - 451.88}{523.88} \times 100 = 13.7 \% \textbf{ decrease.}

Question 9

What is the difference in value of the Barley harvest if sold in dollars during Q1 compared to it being kept and sold in dollars in Q4?

A: $26,876

B: $19,845

C: $19,176

D: $28,350

Written Solutions

Step 1: Calculate the value of the barley harvest in pounds using the correct slice of the pie chart.

27\% of \pounds525,000 \equiv 0.27 \times \pounds525,000

0.27 \times \pounds525,000 = \pounds141,750

Step 2: Convert this value into dollars for Q1 and Q4.

Q1: \pounds1 \equiv \$1.32

\pounds141,750 = 1.32 \times \pounds141,750 = \$187,110

Q4: \pounds1 \equiv \$1.12

\pounds141,750 = 1.12 \times \pounds141,750 = \$158,760 Step 3: Find the difference between the two values.

\$187,110 - \$158,760 = \$28,350

Note: Alternatively, we could have found the difference in the value of the dollar ($0.20) and multiplied this by the value of barley in pounds.

Question 10

If 20% of the potato harvest was damaged over winter, what is the approximate ratio of the value of potatoes sold in dollars in Q2 compared to lettuce sold in Q1?

A: 1: 0.93

B: 1: 1.17

C: 1: 0.85

D: 1: 1.07

Written Solutions

Step 1: Find the initial value of the potato and lettuce harvests in pounds.

Potato:

19% is a multiplier of 0.19, so the initial value is

0.19 \times \pounds525,000 = \pounds99,750

We must subtract 20% for the damaged harvest, this corresponds to a multiplier of 0.800.80 \times \pounds99,750 = \pounds79,800

Lettuce:

17% is a multiplier of 0.17, so the value is

0.17 \times \pounds525,000 = \pounds89,250

Step 2: Calculate the value of potatoes sold in Q2 and lettuce in Q1, in dollars.

Potatoes: \pounds79,800 \times 1.26 = \$100,548

Lettuce: \pounds89,250 \times 1.32 = \$117,810

Step 3: Form the ratio.

\begin{aligned} Potatoes&: Lettuce \\ \$100,548&:\$117,810\end{aligned}

Divide both by the smaller value (potatoes) and evaluate:

\begin{aligned}\dfrac{\$100,548}{\$100,548}&:\dfrac{\$117,810}{\$100,548} \\ 1&:1.17\end{aligned}

Question 11

If the carrot crop is left to grow until Q3 it will sell at 35% higher than the usual price in dollars for this period. What is the difference in expected profit compared to selling in Q1?

A: $8,599.50

B: $4,410.75

C: $18,569.00

D: $9,264.50

Written Solutions

Step 1: Calculate the normal value of the carrot crop in pounds.

6\% \text{of} \pounds525,000 \equiv 0.06 \times \pounds525,000 = \pounds31,500

Step 2: A 35% increase corresponds to a multiplier of 1.35. Use this to calculate the value in pounds of the carrot crop in Q3.

1.35 \times \pounds31,500 = \pounds42,525

Now convert into dollars:

\pounds42,525 \times 1.18 = \$50,179.50

Step 3: Convert the value of carrots sold into dollars for Q1

\pounds31,500 \times 1.32 = \$41,580

Finally find the difference between the two selling prices in dollars

\$50,179.50 - \$41,580 = \bold{\$8,599.50}

Question 12

The value of the dollar is expected to continue to fall at the same rate until Q2 the next year. Additionally, the value of produce will fall by 18% if kept until then. What is the approximate change in the value of StarFarms produce in dollars over the next two quarters?

A: $214k

B: $69k

C: $153k

D: $297k

Written Solutions

Step 1: Calculate the value of the dollar in two quarters time. To do this we first need to find out the percentage change from Q3 to Q4.

\dfrac{Q4 - Q3}{Q3} = \dfrac{\$1.12 - \$1.18}{\$1.18} = -0.051 \text{ or a } 5.1\% \text{ decrease}

Next apply this change twice to the Q4 value to calculate the value of the dollar in Q2 next year. A 5.1% decrease is equivalent to a multiplier of 0.949.

\$1.12 \times 0.949^2 = \$1.01

\text{Note: } 0.949^2 \text{ means to multiply by }0.949\text{ twice}

Step 2: We need to calculate the value of StarFarms produce by next Q2. We are told the value decreases by 18% over the period – this corresponds to a multiplier of 0.82.

\text{Total value of produce }= \pounds525,000

\pounds525,000 \times 0.82 = \pounds430,000

Step 3: Convert the value of the produce to dollars for both Q4 this year and Q2 next year, then find the difference.

\begin{aligned}\text{Q4: }& \pounds525,000 \times \$1.12 = \$588,000 \\ \text{next Q2: }& \pounds430,500 \times \$1.01 = \$434,805 \\ \text{Difference: }& \$588,000 - \$434,805 = \bold{\$153k}\end{aligned}

Question 13

The total amount of Nottingham’s energy supply which is obtained from gas decreased by 6,620,000 kWh from 2011 to 2016. Calculate the percentage increase in total energy usage from 2011 to 2016.

A: 2.1%

B: 1.0%

C: 1.7%

D: 1.3%

Written Solutions

Step 1: Calculate how much gas energy Nottingham used in 2016.

Find the total amount used in 2011 (35% of the total usage, from the graph) and subtract 6.62m from it.

2011: 35\% \text{ of } 1,614m = 0.35 \times 1,614m = 564.9m \text{ kWh of gas energy was used in 2011}

2016: 564.9m - 6.62m = 558.28m \text{ kWh of gas energy used in 2016}

Step 2: Calculate the total energy usage in 2016.

From the graph, we see that gas makes up 34% of total usage in 2016. Therefore, the total energy usage multiplied by 0.34 must equal 558.28m. So, to find the total usage we must divide 558.28m by 0.34.

\text{Total usage } = 558.28m \div 0.34 = 1,642m \text{ kWh}

Step 3: Determine the percentage increase.

\dfrac{1,642m - 1,614m}{1,614m} \times 100 = 1.7\%

Question 14

The costs, per kWh, for nuclear energy in 2001, 2006, and 2011 respectively were $0.26, $0.25, and $0.19. Work out the ratio of the money spent in Nottingham on nuclear energy in 2001 compared to 2006 and 2011.

A: 1.17: 1: 1.03

B: 1.15: 1: 1.04

C: 1.27: 1.13: 1

D: 1.25: 1.18: 1

Written Solutions

Step 1: Work out how much nuclear energy was used in each year.

2001: 88 - 66 = 22\%, \text{ so } 22\% \text{ of } 1,568m = 0.22 \times 1,568m = 344.96m \text{ kWh}

2006: 86 - 66 = 20\%, \text{ so } 20\% \text{ of } 1,592m = 0.20 \times 1,592m = 318.4m \text{ kWh}

2011: 85 - 62 = 23\%, \text{ so } 23\% \text{ of } 1,614m = 0.23 \times 1,614m = 371.22m \text{ kWh}

Step 2: Calculate the money spent on nuclear energy in each year.

\text{2001 cost } = 0.26 \times 344.96m = \$ 89,689,600

\text{2006 cost } = 0.25 \times 318.4m = \$ 79,600,000

\text{2011 cost } = 0.19 \times 371.22m = \$ 70,531,800

Step 3: Form the ratio and simplify by dividing through by the smallest element.

\begin{aligned}\text{2001 cost : 2006 cost : 2011 cost } &= 89,689,600 : 79,600,000 : 70,531,800 \\ &= \dfrac{89,689,600}{70,531,800} : \dfrac{79,600,000}{70,531,800} : \dfrac{70,531,800}{70,531,800} \\ &= 1.27 : 1.13 : 1 \end{aligned}

Question 15

In 2001, the Nottingham council pledged that the total amount, in kWh, of energy that the city obtained from coal would decrease by 10% by 2006. In 2006, they made the same pledge about a 10% decrease between 2006 and 2011. On the occasion when they achieved their goals, by how many kWh did they surpass their goal?
A: 19,969,000 kWh

B: 20,834,000 kWh

C: 21,050,000 kWh

D: 22,716,000 kWh

Written Solutions

Step 1: Calculate how much coal energy was used in 2001, 2006, and 2011.

2001: 66 - 31 = 35\%, \text{ so } 35\% \text{ of } 1,568m = 0.35 \times 1,568m = 548.8m \text { kWh}

2006: 66 - 34 = 32\%, \text{ so } 32\% \text{ of } 1,592m = 0.32 \times 1,592m = 509.44m \text{ kWh}

2011: 62 - 35 = 27\%, \text{ so } 27\% \text{ of } 1,614m = 0.27 \times 1,614m = 435.78m \text{ kWh}

Step 2: Work out what a 10% decrease on the values for 2001 and 2006 are, and check if they are less or greater than the actual values for 2006 and 2011 respectively.

A 10% decrease corresponds to a multiplier of 0.90.

0.90 \times 548.8m = 493.92m < 509.44m \text{ kWh }

0.90 \times 509.44 = 458.496m > 435.78m \text{ kWh } \checkmark

Step 3: We see that the goal was not attained in 2006, but it was in 2011. Work out the difference between the goal amount and actual amount.

458.496m - 435.78m = 22.716m = 22,716,000 \text{ kWh}

Question 16

In 2001, Nottingham’s renewable energy supply is split between wind, solar, and hydroelectric according to the ratio 3:2:1. In 2006, wind made up 65% of the renewable energy supplied to the city. Determine the percentage increase in the amount of wind energy supplied to Nottingham from 2001 to 2006.

A: 43.2%

B: 48.4%

C: 54.0%

D: 59.6 %

Written Solutions

Step 1: Calculate how much renewable energy was used in 2001 and 2006.

2001: 100 - 88 = 12\%, \text{ so } 12\% \text{ of } 1,568m = 0.12 \times 1,568m = 188.16m \text{ kWh}

2006: 100 - 86 = 14\%, \text{ so } 14\% \text{ of } 1,592m = 0.14 \times 1,592m = 222.88m \text{ kWh}

Step 2: Determine how much wind energy was used in 2001 and 2006.

In 2001, the ratio of wind to solar to hydroelectric is 3:2:1. This means that the proportion of the of renewable energy which is provided by wind power is:

\dfrac{3}{3 + 2 + 1} = \dfrac{1}{2} = 50\% of the total amount.

So, we get the following measures for wind energy in 2001 and 2006.

2001: 50\% \text{ of } 188.16m = 0.5 \times 188.16m = 94.08m \text{ kWh}

2006: 65\% \text{ of } 222.88m = 0.65 \times 222.88m = 144.872m \text{ kWh}

Step 3: Work out the percentage increase.

\dfrac{144.872m - 94.08m}{94.08m} \times 100 = 54.0\% \text{ increase}.

Question 17

A tourist leaving York has two options: A) get the train directly to Manchester, or B) get the bus and then take a taxi, which charges £1.40 per mile, from the bus station. Comparing the most expensive bus journey with the least expensive train journey, and assuming the taxi travels at an average of 32mph, work out how long her taxi journey will need to be (to the nearest minute) for option A and B to cost the same amount.

A: 9 minutes

B: 10 minutes

C: 11 minutes

D: 12 minutes

Written Solutions

Step 1: Identify the least expensive train journey and most expensive bus journey and find the difference in cost.

The cheapest train journey is the 09:01 departure, costing £19.70. The most expensive bus journey is the 06:32 departure, costing £11.30.

\text{Difference in cost } = 19.70 - 11.30 = \pounds 8.40

Step 2: Work out how many miles the taxi journey will have to be to cost £8.40.

8.40 \div 1.40 = 6 \text{ miles}

Step 3: Calculate the time taken for this journey.

\begin{aligned}\text{time } &= \text{ distance } \div \text{ speed } \\ \text{journey time } &= 6 \div 32 \\ &= 0.1875 \text{ hours} \end{aligned}

Convert this to minutes, and get: 0.1875 \times 60 = 11 \text{ minutes, to the nearest minute}.

Question 18

If the price of a first-class train ticket is less than 50% greater than the cost of a standard ticket for that journey, then I will opt for the first-class ticket – if it costs more than this, then I will only pay for the standard class. Given that I want to arrive in Manchester before 9am, what is the cost of the cheapest possible journey I will take as a proportion of the most expensive one?

A: 79.8%

B: 76.1%

C: 75.2%

D: 72.6%

Written Solutions

Step 1: Identify which trains before 9am satisfy the criteria that their cost for a first-class ticket is less than 50% greater than the cost of their standard ticket.

A 50% increase corresponds to a multiplier of 1.50.

07:43 arrival: 20.45 \times 1.5 = \pounds 30.68 > \pounds 26.80 \checkmark

08:01 arrival: 20.90 \times 1.5 = \pounds 31.35 > \pounds 31.00 \checkmark

08:18 arrival: 22.50 \times 1.5 = \pounds 33.75 < \pounds 41.00

08:39 arrival: 23.50 \times 1.5 = \pounds 35.25 < \pounds 41.00

Step 2: Work out the cost of the cheapest option as a proportion of the most expensive one.

Given the criteria in the question, the cheapest option is a standard ticket on the 08:18 arrival train which costs £22.50, whilst the most expensive option is the first-class ticket on the 08:01 arrival train which costs £31.00.

As a percentage proportion: \dfrac{22.50}{31.00} \times 100 = 72.6\%.

Question 19

A businessman wishes to travel to Manchester by train and needs to spend 1 hour and 20 minutes on the train preparing for a meeting. When in standard-class, he will always pay for breakfast if it is available. Given these facts, of the journeys he can possibly take, what is the greatest percentage increase from the cost of travelling standard-class to the cost of travelling first-class on the same train?

A: 88.7%

B: 91.2%

C: 79.0%

D: 82.7%

Written Solutions

Step 1: Determine which of the available train journeys the businessman might take.

 The journeys must be at least 1 hour 20 minutes long. We will check each journey’s length by subtracting departure time from arrival time, e.g. 07:43 - 06:15 = 01:28 \checkmark – this we have marked with a tick because it is 80+ minutes long.

08:01 - 06:42 = 01:19

08:18 - 07:02 = 01:16

08:39 - 07:14 = 01:25 \checkmark

09:03 - 07:36 = 01:27 \checkmark

09:05 - 07:47 = 01:18

09:40 - 08:09 = 01:31 \checkmark

10:02 - 08:39 = 01:23 \checkmark

10:20 - 09:01 = 01:19

Step 2: Calculate the percentage increase from the cost of a standard ticket to the cost of a first-class ticket for each of the 5 identified journeys.

Breakfast is available on two of the relevant journeys, the 7:14 departure and the 8:09 departure. We must add £5 on to the cost of the standard-class ticket in each of these cases.

06:15 departure: \dfrac{26.80 - 20.45}{20.45} \times 100 = 31.1\% \text{ increase}

07:14 departure: \dfrac{41.00 - (23.50 + 5)}{23.50 + 5} \times 100 = 43.9\% \text{ increase}

07:36 departure: \dfrac{48.60 - 25.75}{25.75} \times 100 = 88.7\% \text{ increase}

08:09 departure: \dfrac{47.25 - (29.00 + 5)}{29.00 + 5} \times 100 = 39.0\% \text{ increase}

08:39 departure: \dfrac{35.10 - 23.20}{23.20} \times 100 = 51.3\% \text{ increase}

Step 3: Identify the greatest percentage increase to be 88.7% on the 07:36 train.

Question 20

Person A will use the Wi-fi on their bus for the full-length of the journey, however Person B will only use it for the first 1.5 hours. What is the ratio of the cost of a ticket for Person A on the longest available bus journey, compared to a ticket for Person B on the shortest available bus journey?

A: 1.18: 1

B: 1.23: 1

C: 1: 1.07

D: 1: 1.24

Written Solutions

Step 1: Identify which bus journey is shortest and which is longest.

Subtract each departure time from the arrival time, e.g. 08:58 - 06:10 = 02:48.

09:15 - 06:32 = 02:43

09:35 - 07:02 = 02:33

10:04 - 07:39 = 02:25

10:26 - 07:59 = 02:27

The longest journey is the 06:10 departure and the shortest is the 07:39 departure.

Step 2: Work out the cost of these journeys with wi-fi use considered.

Person A takes the longest journey and will use wi-fi for its entire length. The journey is 2 hours and 48 minutes long, meaning the first 2 hours will cost £2, and the latter 48 minutes will cost 48 \times 0.03 = \pounds1.44. So, the total cost of this journey is

10.30 + 2 + 1.44 = \pounds 13.74.

Person B takes the shortest journey and will use 1.5 hours of wi-fi, costing an extra £2, so the total cost of this journey will be

9.60 + 2 = \pounds 11.60.

Step 3: Form the ratio and simplify by dividing through by the smallest element.

\begin{aligned}\text{Person A on longest : Person B on shortest } &= 13.74 : 11.60 \\ &= \dfrac{13.74}{11.60} : \dfrac{11.60}{11.60} \\ &= 1.18 : 1 \end{aligned}