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Worked Solutions – Test 3 2018-11-01T17:45:42+00:00

## Worked Solutions – Test 3

#### Question 1

The monthly prices of Alola and Alola (EU) are representative of the pounds to euro exchange rate. What is the approximate price of a new 18-month Nile subscription in euros?

A: £121

B: £171

C: £193

D: £142

#### Written Solutions

Step 1: First we must calculate the euro exchange rate. We do this by writing down the ratio between the monthly prices for Alola and Alola (EU) and simplifying.

\begin{aligned}\pounds10.49 &: 12.49\text{ euros} \\ \dfrac{10.49}{10.49} &: \dfrac{12.49}{10.49} \\ \pounds1 &: 1.19\text{ euros} \end{aligned}

Step 2: Next calculate the total cost of an 18-month Nile subscription in pounds. The first two months are free so 16 months are to be paid for.

$16 \times\pounds8.99 = \pounds143.84$

Step 3: Finally convert this into euros using the exchange rate we calculated earlier.

$\pounds143.84 \times 1.19 \text{ euros} \approx \pounds171$

#### Question 2

Netview (US) gains 15m new subscribers over 12 months, of which 23% opt for the additional Super HD upgrade. What is the percentage change in profit for Netview (US), assuming costs stay constant?

A: 18.2%

B: 16.8%

C: 16.3%

D: 14.2%

#### Written Solutions

Step 1: Calculate the increase in revenue for Netview (US).

$15\text{m} \times \13.49 = \202.35\text{m (basic monthly revenue from the new subscribers)}$

These new subscribers each get a months free trial, so over 12 months they pay for 11 months each.

$\202.35\text{m} \times 11 = \2,225.85\text{m}$

Additionally, we need to calculate the value of those who opt for Super HD. This is over 11 months, but first we need to calculate 23% of 15m.

$23\% \text{ of } 15\text{m} = 0.23 \times 15,000,000 = 3.45\text{m}$

$3.45\text{m} \times 11 \times \5.00 = \189.75\text{m}$

So, the total increase in revenue is:

$\189.75\text{m} + \2,225.85\text{m} = \2,415.6\text{m}$

Step 2: Calculate the original and new profit.

\begin{aligned}\text{Original: }&\16,510\text{m} - \3,250\text{m} = \13,260\text{m} \\ \text{New: }&\16,510\text{m} + \2,415.6\text{m} - \3,250\text{m} = \15,675.6\text{m}\end{aligned}

Step 3: Work out the percentage change.

$\dfrac{\15,675.6\text{m} - \13,260\text{m}}{\13,260\text{m}} \times 100 = \bold{18.2\%}$

#### Question 3

Yearly revenues fall by 12% for all three UK services due to established customers cancelling their subscriptions. Assuming 34% of users had the Super HD package (where available), what is the approximate total number of customers leaving UK services?

A: 11.2m

B: 10.4m

C: 8.4m

D: 7.6m

#### Written Solutions

Step 1: Calculate the fall in revenue for each company. We are not concerned about the new revenue, just the decrease of 12% so we use a multiplier of 0.12.

\begin{aligned}\text{Netview: }&\pounds4,945\text{m} \times 0.12 = \pounds593.4\text{m} \\ \text{Nile: }&\pounds3,990\text{m} \times 0.12 = \pounds478.8\text{m} \\ \text{Alola: }&\pounds2,640\text{m} \times 0.12 = \pounds316.8\text{m}\end{aligned}

Step 2: Now we need to calculate what an average user spends. Each user pays the monthly fee and 34% of users get Super HD. So, we can say the average user pays monthly and pays 34% (a multiplier of 0.34) of the cost of Super HD. We need to calculate this for each UK service.

\begin{aligned}\text{Netview: }&(\pounds9.99 + (0.34 \times \pounds4.00)) \times 12\text{ months} = \pounds136.20 \text{ over 12 months} \\ \text{Nile: }&(\pounds8.99 + (0.34 \times \pounds5.00)) \times 12\text{ months} = \pounds128.28 \\ \text{Alola: }&(\pounds10.49 + (0.34 \times \pounds3.00)) \times 12\text{ months} = \pounds138.12\end{aligned}

Step 3: Finally divide the fall in revenue by the average yearly spend for each service and add up the total.

\begin{aligned}\text{Netview: }&\pounds593.4\text{m} \div \pounds136.20 \approx 4.4\text{m} \\ \text{Nile:}&\pounds478.8\text{m} \div \pounds128.28 \approx 3.7\text{m} \\ \text{Alola: }&\pounds316.8\text{m} \div \pounds138.12 \approx 2.3\text{m}\end{aligned}

$4.4\text{m} + 3.7\text{m} + 2.3\text{m} = \bold{10.4\text{m lost subscribers}}$

#### Question 4

What is the highest average profit per user across all services in GBP? Assume that the price of Netview in the UK and US is set to match the value of the dollar.

A: £109.66

B: £130.00

C: £100.28

D: £102.38

#### Written Solutions

Step 1: Calculate the average profit per user in their original currencies. We do this by finding the difference between revenue and costs, then dividing by the total number of users.

\begin{aligned}\text{Netview: }&\dfrac{\pounds4,945\text{m} - \pounds870\text{m}}{45\text{m}} = \pounds90.56 \\ \text{Netview (US): }&\dfrac{\16,510\text{m} - \3,250\text{m}}{102\text{m}} = \130.00 \\ \text{Nile: }&\dfrac{\pounds3,990\text{m} - \pounds1,540\text{m}}{37\text{m}} = \pounds66.22 \\ \text{Alola: }&\dfrac{\pounds2,640\text{m} - \pounds490\text{m}}{21\text{m}} = \pounds102.38 \\ \text{Alola (EU): }&\dfrac{3,980\text{m euros} - 800\text{m euros}}{29\text{m}} = 109.66\text{ euros} \end{aligned}

Step 2: Using the monthly price of Netview in the UK and US, calculate the pounds to dollar exchange rate.

\begin{aligned}\pounds9.99 &: \13.49 \\ \dfrac{9.99}{9.99} &: \dfrac{13.49}{9.99} \\ 1 &: 1.35\end{aligned}

Step 3: Convert the values for Netview (US) and Alola (EU) into GBP. We know £1 = $1.35 and$1 = €0.81.

\begin{aligned}\text{Netview (US): }&\130 \div 1.35 = \pounds96.30 \\ \text{Alola (EU): }&109.66\text{ euros} \div 0.81 = \135.38 \\ &\135.38 \div 1.35 = \pounds100.28\end{aligned}

Clearly, Alola (UK) has the highest average profit per customer at £102.38.

#### Question 5

A businesswoman has meetings at Embankment, Westminster and Baker Street on a Tuesday that will last for 1 hour, 70 minutes and 1:05 respectively. She will also take a break after her third meeting for 20 minutes. Assuming she leaves at 06:26 from Kings Cross and only takes the underground, what is the earliest time she will arrive back?

A: 11:21

B: 11:26

C: 12:41

D: 12:46

#### Written Solutions

She departs Kings cross at 06:26, on the first repeat of our timetable (75 minutes later). We will follow her journey through as it goes.

Arrives at Embankment at 06:49 and has an hour-long meeting until 07:49. The next train will come 75 minutes after the arrival at 06:49 so at $06\colon49 + 01\colon15 = 08\colon04$.

The 08:04 train from Embankment arrives at Westminster at 08:07, and her 70-minute meeting lasts until 09:17. The next train arrives 75 minutes after the 08:07 at 09:22.

The 09:22 arrives at Baker Street at 09:53 and her 1:05 meeting lasts until 10:58, with her break lasting until 11:18. Trains arrive at Baker Street in 75-minute intervals, so the next train from 09:53 is the 11:08, and after that the 12:23.

She takes the 12:23 back to Kings Cross, which arrives at 12:41.

#### Question 6

Black Cab and Zooms run the route from Kings Cross to Liverpool Street for 6 hours from 09:30. Assuming each journey normally lasts 10 minutes, what is the difference in profit for the day between the two companies?

A: £30.63

B: £27.75

C: £31.86

D: £29.40

#### Written Solutions

Step 1: Convert the price per km for Black Cab into miles.

$\pounds1.40\text{ per km} \div 1.6 = \pounds0.88\text{ per mile}$

Step 2: 6 hours from 09:30 is 15:30, meaning the journeys do not take place in rush hour. 6 hours divided by 10 minutes is 60 journeys over the period.

We need to know how much profit each journey makes, which means we need to know the distance travelled. From the table the distance between Kings Cross and Liverpool Street is 3.3 miles.

Calculate how much profit each journey makes for the two companies. We must multiply the rate per mile by 0.82 for an 18% reduction due to tax.

\begin{aligned}\text{Black Cab: }&\pounds2.40 + (\pounds0.88 \times 3.3\text{ miles} \times 0.82) = \pounds4.78 \\ \text{Zooms: }&\pounds2.70 + (\pounds0.95 \times 3.3\text{ miles} \times 0.82) = \pounds5.27\end{aligned}

Step 3: Finally calculate the total profit for each company over the day and find the difference.

\begin{aligned}\text{Black Cab: }&\pounds4.78 \times 60 = \pounds286.80 \\ \text{Zooms: }&\pounds5.27 \times 60 = \pounds316.20 \\ \\ \text{Difference: }&\pounds316.20 - \pounds286.80 = \bold{\pounds29.40}\end{aligned}

#### Question 7

What is the total cost of a journey for four people on a Wednesday from Kings Cross on the 07:41 train to Embankment, then by Royal taxi to Notting Hill, and finally back to Kings Cross on the next train?

A: £41.60

B: £42.35

C: £44.55

D: £46.85

#### Written Solutions

Step 1:  Calculate the total cost and duration of the first leg of the journey.

A single from Kings Cross to Embankment costs $\pounds1.20 + \pounds1.32 + \pounds0.60 = \pounds3.12$ so the cost for four people is $\pounds3.12 \times 4 = \pounds12.48$. The trains arrive in 75 minute intervals on a Wednesday and the 07:41 is 02:30 later than our timetable, so it arrives in Embankment at 08:04.

Step 2: Calculate the cost of the taxi from Embankment to Notting Hill.

Royal taxis cost £3.10 and then £1.70 per km, which is $\pounds1.70 \div 1.61 = \pounds1.06\text{ per mile}$. However, departing Embankment at 08:04 means it is rush hour, and there will be a 40% increase in cost per mile. So this is a multiplier of 1.4. From the table it is 8.9 miles to Notting Hill.

$\pounds3.10 + (\pounds1.06 \times8.9\text{ miles} \times 1.4) = \pounds16.31$

We do not multiply by 4 as the taxi is shared.

Step 3: Calculate the cost of the train from Notting Hill to Kings Cross.

Trains are the same cost regardless of the time of day, so the cost of the train is simply $\pounds1.34 + \pounds0.96 + \pounds1.64 = \pounds3.94\text{ each}$

So for all four people the cost is $\pounds3.94 \times 4 = \pounds15.76$

Step 4: Finally add these three steps up to get a total cost of £44.55.

#### Question 8

Royal and Elite both travel at an average of 28mph, whereas other taxis travel at 30mph. What is the cost of the cheapest taxi that makes the journey from Westminster to Kings Cross faster than the train at 07:37 on a Saturday?

A: £21.57

B: £23.23

C: £25.44

D: £24.90

#### Written Solutions

Step 1: Figure out how long it takes the slower and faster taxis to make the journey.

It is $7.6 + 4.5 + 1.9 + 3.1 = 17.1$ miles from Westminster to Kings Cross. We need to calculate the time taken at 28mph and 30mph so we use the following:

$\text{Time } = \dfrac{\text{Distance}}{\text{Speed}}$

$\text{Duration at 28mph } = \dfrac{17.1}{28} \times 60 = 36.64\text{ minutes}$

$\text{Duration at 30mph } = \dfrac{17.1}{30} \times 60 = 34.2\text{ minutes}$

Since it is rush hour we must increase these durations by 40%, or a multiplier of 1.4.

$36.64\text{ minutes} \times 1.4 = 51.3\text{ minutes}$

$34.2\text{ minutes} \times 1.4 = 47.88\text{ minutes}$

Compared to the time taken by the train (49 minutes) this means the 28mph taxis will not make it in time.

Step 2: Now we need to compare the costs of the three fastest taxis. We first need to convert the costs per km to miles where needed by dividing by 1.61.

$\pounds1.40 \div 1.61 = \pounds0.87 \text{ and } \pounds1.80 \div 1.61 = \pounds1.12$

Calculate the cost of each taxi, considering that the cost per mile must be increased by 40% or a multiplier of 1.4.

\begin{aligned}\text{Black Cab: }&\pounds2.40 + (1.4 \times \pounds0.87 \times 17.1) = \bold{\pounds23.23} \checkmark \\ \text{Zooms: }&\pounds2.70 + (1.4 \times \pounds0.95 \times 17.1) = \pounds25.44 \\ \text{Kings: }&\pounds3.60 + (1.4 \times \pounds1.12 \times 17.1) = \pounds30.41\end{aligned}

#### Question 9

Next May the number of days of rainfall in each city is expected to increase by 25% YoY, while the average rainfall per day will increase by 30%. What is an estimate for the total rainfall for all three cities combined next May?

A: 9.8 inches

B: 12.1 inches

C: 10.4 inches

D: 11.8 inches

#### Written Solutions

Step 1: Calculate the current average rainfall per day by dividing the total rainfall by the number of days for each city.

\begin{aligned}\text{London: }&1.8 \div 11 = 0.16\text{ inches per day} \\ \text{Birmingham: }&2.4 \div 14 = 0.17\text{ inches per day} \\ \text{Manchester: }&2.2 \div 17 = 0.13\text{ inches per day} \end{aligned}

Step 2: Increase the number of days by 25% (x1.25) and the average rainfall by 30% (x1.3).

Next May (days):

\begin{aligned}\text{London: }&12 \times 1.25 = 15 \\ \text{Birmingham: }&16 \times 1.25 = 20 \\ \text{Manchester: }&20 \times1.25 = 25 \end{aligned}

Next May (average rainfall per day):

\begin{aligned}\text{London: }&0.16 \times 1.3 = 0.21 \\ \text{Birmingham: }&0.17 \times 1.3 = 0.22 \\ \text{Manchester: }&0.13 \times 1.3 = 0.17 \end{aligned}

Step 3: Multiply the number of days next May by the average rainfall per day to get the total rainfall for each city, then add them up.

\begin{aligned}15 \times 0.21 &= 3.15 \\ 20 \times0.22 &= 4.4 \\ 25 \times 0.1677 &= 4.25 \\ 3.15 + 4.4 + 4.25 &\approx \bold{11.8\text{ inches}}. \end{aligned}

#### Question 10

What is the sum of the average rainfalls per day in April and May for each city, expressed as a ratio.

A: 1.52: 1.30: 1

B: 1.24: 1: 1.08

C: 1.46: 1.10: 1

D: 1.37: 1: 1.21

#### Written Solutions

Step 1: Calculate the average rainfall in April and May for each city.

\begin{aligned}\text{London: April: }&1.8 \div 7 \approx 0.26 \\ \text{May: }& 1.8 \div 12 = 0.15 \\ \text{Birmingham: April: }&2 \div 10 = 0.2 \\ \text{May: }& 2.4 \div 16 = 0.15 \\\text{Manchester: April: }&2.1 \div 13 \approx 0.16 \\ \text{May: }& 2.2 \div 20 = 0.11 \end{aligned}

Step 2: Combine for each city.

\begin{aligned}\text{London: }&0.26 + 0.15 = 0.41 \\ \text{Birmingham: }&0.2 + 0.15 = 0.35 \\ \text{Manchester: }&0.16 + 0.11 = 0.27 \end{aligned}

Step 3: Form the ratio and simplify.

$\begin{gathered}\text{London: Birmingham: Manchester} \\ 0.41 : 0.35 : 0.27 \\ \dfrac{0.41}{0.27} : \dfrac{0.35}{0.27}: \dfrac{0.27}{0.27} \\ \bold{1.52 : 1.30 : 1} \end{gathered}$

#### Question 11

Based on the percentage decrease in total rainfall from January to February in each city, what was an estimate for the total rainfall in December in all three cities combined?

A: 9.8 inches

B: 9.6 inches

C: 10.2 inches

D: 10.4 inches

#### Written Solutions

Step 1: Calculate the percentage decrease in total rainfall from January to February for each city.

\begin{aligned}\text{London: }&\dfrac{1.5 - 1.90}{1.90} \times 100 \approx -21.1\% \\ \text{Birmingham: }&\dfrac{1.6 - 2.3}{2.3} \times100 \approx -30.4\% \\ \text{Manchester: }& \dfrac{2 - 2.8}{2.8} \times 100 \approx -28.6\% \end{aligned}

Step 2: Reverse this percentage decrease by dividing the January figures by the original multiplier to find the total rainfall for each city in December. I.e. a 21.1% decrease is a multiplier of 0.789 so divide by 0.789.

\begin{aligned}\text{London: }&1.9 \div 0.789 \approx 2.41 \\ \text{Birmingham: }&2.3 \div 0.696 \approx 3.30 \\ \text{Manchester: }&2.8 \div 0.714 \approx 3.92 \end{aligned}

Step 3: Add up these figures to calculate an estimate for the total rainfall in December.

$2.41 + 3.30 + 3.92 \approx 9.6\text{ inches}$

#### Question 12

12 days of the minimum rainfall to require recording were missed in Birmingham in each of January, February and March. What is the difference in the proportion of Birmingham’s rainfall (compared to the total for all three cities) before this data was added and after?

A: 5.0%

B: 4.5%

C: 4.0%

D: 5.5%

#### Written Solutions

Step 1: Calculate the original proportion of rainfall attributed to Birmingham.

$\text{Total combined rainfall: } 1.9 + 2.3 + 2.8 + 1.5 + 1.6 + 2 + 1.6 + 1.9 + 2.4 = 18\text{ inches}$

$\text{Original Propotion } = \dfrac{2.3 + 1.6 + 1.9}{18} \times 100 \approx 32.2\%$

Step 2: Calculate the new total rainfall for Birmingham, and for all cities from January – March.

We have $0.04 \times 12\text{ days } \times 3\text{ months} = 1.44\text{ inches}$ to add on to the total rainfall, and to Birminghams rainfall. Now we repeat the earlier calculation for proportion:

$\text{New Propotion } = \dfrac{2.3 + 1.6 + 1.9 + 1.44}{18 + 1.44} \times 100 \approx 37.2\%$

Step 3: The difference in proportion is 37.2% – 32.2% = 5.0%

#### Question 13

Express the combined GDP of Los Angeles and New York in 2017 as a ratio compared to 2018 and 2019.

A: 1.06: 1.05: 1

B: 1.12: 1.08: 1

C: 1.09: 1.07: 1

D: 1.03: 1: 1.05

#### Written Solutions

Step 1: Calculate the combined share of GDP for Los Angeles and New York in 2017, 2018 and 2019

$\begin{gathered}2017: 2018: 2019 \\ 26\% + 30\%: 24\% + 30\%: 22\% + 28\% \\ 56\%: 54\%: 50\% \end{gathered}$

Step 2: Calculate the combined share of GDP for each year.

$\begin{gathered}56\%\text{ of }\3,430\text{bn}: 54\%\text{ of }\3,520\text{bn}: 50\%\text{ of }\3,625\text{bn} \\ 0.56 \times 3430: 0.54 \times 3520: 0.5 \times 3625 \\ \1,920.8\text{bn}: \1,900.8\text{bn}: \1,812.5\text{bn}\end{gathered}$

Step 3: Simplify the ratio by dividing through by the lowest element.

$\begin{gathered}\1,920.8\text{bn}: \1,900.8\text{bn}: \1,812.5\text{bn} \\ \dfrac{1920.8}{1812.5}: \dfrac{1900.8}{1812.5}: \dfrac{1812.5}{1812.5} \\ \bold{1.06: 1.05: 1} \end{gathered}$

The GDP of the next wealthiest city in 2016, Seoul, was $780bn. This then increased by 5% year-on-year. If added to the data in 2019, what would the GDP of Seoul be as a proportion of the combined GDP of the four cities? A: 18.5% B: 18.7% C: 19.8% D: 19.9% #### Written Solutions Step 1: Find out what the GDP of Seoul is in 2019 by increasing the GDP in 2016 by 5% three times. This increase corresponds to a multiplier of 1.05. $\780\text{bn } \times 1.05^3 \approx \903\text{bn}$ Note: $1.05^3 \text{ is shorthand for }1.05 \times 1.05 \times 1.05$ Step 2: We need to work out the new combined GDP of the four cities in 2019. $\3,625\text{bn } + \903\text{bn } = \4,528\text{bn}$ Step 3: Finally, calculate Seoul’s GDP as a proportion of the total. $\dfrac{\903\text{bn}}{\4,528\text{bn}} \times 100 = \bold{19.9\%}$ #### Video Solutions #### Question 15 From 2019 to 2020 the GDP of Los Angeles, New York and Tokyo increased by 7%,$72bn and 102bn respectively. What are the proportions of combined GDP for each of the cities in 2020? A: 20%, 31% and 49% B: 21%, 27% and 52% C: 22%, 28% and 50% D: 19%, 29% and 52% #### Written Solutions Step 1: First work out the GDP for each city in 2019. \begin{aligned}\text{Los Angeles: }&22\%\text{ of }\3,625\text{bn} = 0.22 \times 3625 = \797.5\text{bn} \\ \text{New York: }&28\%\text{ of }\3,625\text{bn} = 0.28 \times 3625 = \1,015\text{bn} \\ \text{Tokyo: }&50\%\text{ of }\3,625\text{bn} = 0.50 \times 3625 = \1,812.5\text{bn}\end{aligned} Step 2: Next work out the new GDP by either increasing by a multiplier (7% = 1.07) or adding on the increase. \begin{aligned}\text{Los Angeles: }&\797.5\text{bn} \times 1.07 = \853.3\text{bn} \\ \text{New York: }&\1,015\text{bn} + \72\text{bn} = \1,087\text{bn} \\ \text{Tokyo: }&\1,812\text{bn} + \102\text{bn} = \1,914\text{bn}\end{aligned} Step 3: Calculate the total GDP in 2020 and use this to calculate the proportion of GDP attributed to each city. $\text{Combined GDP in 2020 } = \853.3\text{bn } + \1,087\text{bn } + \1,914\text{bn } = \3,854.3\text{bn}$ \begin{aligned}\text{Los Angeles: }&\dfrac{\853.3\text{bn}}{\3,854.3\text{bn}} \times 100 \approx \bold{22\%} \\ \text{New York: }&\dfrac{\1,087\text{bn}}{\3,854.3\text{bn}} \times 100 \approx \bold{28\%} \\ \text{Tokyo: }&\dfrac{\1,914\text{bn}}{\3,854.3\text{bn}} \times 100 \approx \bold{50\%}\end{aligned} #### Video Solutions #### Question 16 What was the highest increase in value of any city in one year over the period? A:169.2bn

B: $193.3bn C:$202.5bn

D: \$176.1bn

#### Written Solutions

Step 1: Eliminate cities or dates we do not need to consider.

By examining the chart, we can tell that the percentage share of Los Angeles decreases each year. While the value may still increase (since Combined GDP increases) we are looking for a city whose percentage share increases also.

Tokyo is the only city that does this, as New York’s share also decreases year-on-year.

Step 2: Calculate the GDP of Tokyo for each year.

$\begin{array}{c|c} 2016 & 40\%\text{ of }\3,350\text{bn } = 0.4 \times \3,350\text{bn } = \1,340\text{bn} \\ 2017 & 44\%\text{ of }\3,430\text{bn } = 0.44 \times \3,430\text{bn } = \1,509.2\text{bn} \\ 2018 & 46\%\text{ of }\3,520\text{bn } = 0.46 \times \3,520\text{bn } = \1,619.2\text{bn} \\ 2019 & 50\%\text{ of }\3,625\text{bn } = 0.5 \times \3,625\text{bn } = \1,812.5\text{bn}\end{array}$

Step 3: compare the figures to find the largest increase in value in one year.

\begin{aligned}\text{2017 - 2016 }& = \1,509.2\text{bn} - \1,340\text{bn} = \169.2\text{bn} \\ \text{2018 - 2017 }& = \1,619.2\text{bn} - \1,509.2\text{bn} = \110\text{bn} \\ \text{2019 - 2018 }& = \1,812.5\text{bn} - \1,619.2\text{bn} = \bold{\193.3\text{bn}}\end{aligned}

#### Question 17

What was the percentage change in voter turnout over the period, assuming a voting population of 60m in 2005, that increased by 24% to 2017?

A: 3.0% increase

B: 3.3% increase

C: 2.1% decrease

D: 4.6% decrease

#### Written Solutions

Step 1:  Calculate voter turnout in 2005. We need to know the total number of votes cast and write that as a proportion of the voting population.

$3 + 6 + 9 + 9.5 = 27.5\text{ million votes in 2005}$

$\dfrac{27.5\text{m}}{60\text{m}} \times 100 = 45.8\%$

Step 2: Calculate voter turnout in 2017, considering the voting population increase of 24%. This is a multiplier of 1.24 so we multiply 60m by 1.24.

$2.5 + 3.5 + 13 + 13.5 = 32.5\text{ million votes in 2010}$

$\dfrac{32.5\text{m}}{60\text{m } \times 1.24} \times 100 = 43.7\%$

Step 3: Finally, we need to work out the percentage change in voter turnout between 2005 and 2017.

$\dfrac{43.7 - 45.8}{45.8} \times 100 = \bold{4.6\%\text{ decrease}}$

#### Question 18

People aged 25 – 50 and 50+ vote in the ratio 4:7. What is the ratio of votes for LAB in 2017 from under 24s, 25-50 and 50+?

A: 1: 1.34: 2.12

B: 1: 1.29: 2.26

C: 1: 1.14: 1.96

D: 1: 1.48: 2.37

#### Written Solutions

Step 1: First we need to know the vote from under 24s in 2017. To do this we first need to calculate the vote in 2010 and increase it by 450,000 votes for each subsequent election (two).

$23\%\text{ of } 8.5\text{m} = 0.23 \times 8.5 = 1.955\text{million votes}$

Now we increase by 450,000 twice:

$1.955\text{m} + 450,000 + 450,000 = 2.855\text{million votes in 2017}$

Step 2: Calculate the actual voting numbers for the other two age brackets. To do this we first need to subtract the votes from under 24s to get the total vote to be split between 25-50s and 50+.

$13\text{m} - 2.855\text{m} = 10.145\text{m}$

The ratio is 4: 7, meaning there are 11 shares in total. Divide the total number of votes left by 11 to determine the value of one share.

$10.145\text{m} \div 11 = 0.92\text{m}$

Now multiply the ratio by this amount to determine the number of votes for each age group.

\begin{aligned}\text{25 to 50}&:\text{50+} \\ 4&: 7 \\ 4 \times 0.92\text{m}&: 7 \times 0.92\text{m} \\ 3.68\text{m}&: 6.44\text{m}\end{aligned}

Step 3: Include the under 24 bracket in this ratio (as they are now in the same units) and simplify.

$\begin{gathered}\text{Under 24s}:\text{25 to 50}:\text{50+} \\ 2.855: 3.68: 6.44 \\ \dfrac{2.855}{2.855}: \dfrac{3.68}{2.855}: \dfrac{6.44}{2.855} \\ \bold{1: 1.29: 2.26} \end{gathered}$

#### Question 19

36% of the vote is required to form a government. In how many of the years was this passed by at least one party?

A: 1

B: 2

C: 3

D: 4

#### Written Solutions

Step 1: First calculate the total vote in each year.

\begin{aligned}\text{2005: }&3 + 6 + 9 + 9.5 = 27.5\text{m} \\ \text{2010: }&3.5 + 7 + 11 + 8.5 = 30\text{m} \\ \text{2015: }&8 + 2.5 + 11 + 9.5 = 31\text{m} \\ \text{2017: }&3.5 + 2.5 + 13.5 + 13 = 32.5\text{m}\end{aligned}

Step 2: Now we need to calculate the proportion of the vote for each party with the highest vote in each year. We do not need to calculate the others as we are only concerned with years in which at least one party passed the 36% threshold.

\begin{aligned}\text{2005 LAB: }&\dfrac{9.5}{27.5} \times 100 = 34.5\% \\ \text{2010 CON: }&\dfrac{11}{30} \times 100 = 36.7\% \checkmark \\ \text{2015 CON: }&\dfrac{11}{31} \times 100 = 35.5\% \\ \text{2017 CON: }&\dfrac{13.5}{32.5} \times 100 = 41.5\% \checkmark\end{aligned}

This occurs twice over the period.

#### Question 20

The Other category is comprised of four other parties, A, B, C and D. A and B receive votes in the ratio 1:2, C and D receive votes in the ratio 4:5 and C receives double the votes of A. From 2005 to 2017, what was the total number of votes for A and D combined?

A: 8.4m

B: 7.6m

C: 8.1m

D: 7.9m

#### Written Solutions

Step 1: We need to figure out the ratio of A: B: C: D from the information given. We are given that:

$\begin{array}{c|c} A : B & 1 : 2 \\ C: D & 4: 5 \\ A : C & 1 : 2\end{array}$

In order to combine ratios, we must write them so the shares for a single party are similar. To combine Ratio 1 and 3, we need to write part A as the same number of shares, which it already is. We then write:

$\begin{array}{c|c} A : B : C& 1 : 2 : 2 \\ C: D & 4: 5 \end{array}$

Now multiply the ratio for A: B: C by 2 so that the above shares for C are similar.

$\begin{array}{c|c} A : B : C& 2 : 4 : 4 \\ C: D & 4: 5 \end{array}$

We can then combine these ratios to form a ratio for all four

$\begin{array}{c|c} A : B : C: D& 2 : 4 : 4 : 5 \end{array}$

Step 2: Now that we have this, we notice that A and D combined have 7 shares out of a total of 15. If we split the total number of votes for the period into 15 parts and multiply by the 7 we want, we will have our answer. So the total number of votes for the Other category from 2005 to 2017 is:

$3 + 3.5 + 8 + 3.5 = 18\text{m}$

$\dfrac{18\text{m}}{15} \times 7 = \bold{8.4\text{ million combined votes for A and D}}$