## Worked Solutions – Test 4

**Question 1**

__Q1__

What is the percentage decrease in total gross income for a branch consisting of 1 senior manager, 2 managers and 6 customer service representatives from 2010 to 2020?

A: 7.08%

B: 6.24%

**C: 7.48%**

D: 6.87%

**Written Solutions**

**Step 1:** Calculate the total gross income in 2010.

\pounds34,000 + (\pounds27,000 \times 2\text{ managers}) + (\pounds21,000 \times 6\text{ customer service reps}) = \pounds214,000

**Step 2:** Calculate the total gross income in 2020.

\pounds26,000 + (\pounds26,000 \times 2\text{ managers}) + (\pounds20,000 \times 6\text{ customer service reps}) = \pounds198,000

**Step 3:** Calculate the percentage decrease as follows:

\dfrac{\pounds198,000 - \pounds214,000}{\pounds214,000} \times100 = \bold{7.48\%\text{ decrease}}

**Video Solutions**

**Question 2**

What is the ratio of total pension contributions per year for a new senior manager in 2010 compared to the total contributions per year after being with the company for 10 years?

A: 1: 1.03

B: 1: 1.04

**C: 1: 1.05**

D: 1: 1.06

**Written Solutions**

**Step 1:** Find out the initial pension contributions in 2010, based on 1% employee contribution and 3% employer contribution.

We multiply by 0.01 and 0.03 for 1% and 3% respectively.

\pounds34,000 \times (0.01 + 0.03) = \pounds1,360

**Step 2:** 10 years is three full sets of three years, which equated to a new employee contribution of 2.5%. Calculate this based on the figures in 2020.

\pounds26,000 \times (0.025 + 0.03) = \pounds1,430

**Step 3:** Form the ratio as required and simplify.

\begin{aligned}2010 &: 2020 \\ \pounds1,360 &: \pounds1,430 \\ \dfrac{1360}{1360} &: \dfrac{1430}{1360} \\ 1 &: 1.05\end{aligned}

**Video Solutions**

**Question 3**

What proportion of gross income in 2015 is deducted due to income tax and pension contributions for a manager who has been at the firm for 20 years?

A: 12.4%

**B: 14.4%**

C: 17.4%

D: 18.4%

**Written Solutions**

**Step 1:** Determine the employee contributions of 3% in 2015.

3\%\text{ of }\pounds28,000 \equiv 0.03 \times \pounds28,000 = \pounds840

**Step 2:** Calculate the tax paid on £28,000. This is in the third tax band so we need to do a number of calculations.

\pounds28,000 - \pounds20,000 = \pounds8,000

This is taxed at 24% so we need to calculate 24% of £8,000 or a multiplier of 0.24.

0.24 \times \pounds8,000 = \pounds1,920

Now we calculate the tax in the next band, £11.5k – 20k.

\pounds20,000 - \pounds11,500 = \pounds8,500

We need 15% of this value which is a multiplier of 0.15.

0.15 \times \pounds8,500 = \pounds1,275

Finally, there is no tax paid on the value below £11,500 so the total tax paid is \pounds1,920 + \pounds1,275 = \pounds3,195

**Step 3:** The total outgoings are \pounds840 + \pounds3,195 = \pounds4,035 so we need to calculate this as a percentage of the total gross income for the year.

\dfrac{\pounds4,035}{\pounds28,000} \times 100 = \bold{14.4\%}

**Video Solutions**

**Question 4**

What is the yearly gross income minus pensions contributions for a new customer service representative in 2020 if pension contributions do not start until after three months of employment?

** A: £19,850**

B: £19,800

C: £19,875

D: £19,825

**Written Solutions**

**Step 1:** Three months without pensions contributions means that for three months they get their normal wage, which is £20,000 shared over 12 months.

\pounds20,000 \div 12 \approx \pounds1,666.67

** Step 2:** For nine months of the year they must pay pensions contributions. First calculate their wage over nine months.

\pounds1,666.67 \times 9 \approx \pounds15,000

Now calculate the pension contributions on this. For a new employee this it 1% or a multiplier of 0.01.

0.01 \times \pounds15,000 = \pounds150

** Step 3:** So over the course of the year they can expect to have \pounds20,000 - \pounds150 = \bold{\pounds19,850} in gross income after their pension contributions are taken off.

**Video Solutions**

**Question 5**

What was the percentage increase in the value of oil outputted by all three countries combined between 1988 and 2018? Assume 365 days in one year.

A: 79.4%

B: 83.7%

C: 76.2%

**D: 81.8%**

**Written Solutions**

**Step 1:** Calculate the amount of oil produced in both 1988 and 2018.

\begin{aligned}\text{1988: }& 9,700 + 10,100 + 10,400 = 30,200 \text{ (000 barrels per day)} \\ &30.2\text{m} \times 365 = 11,023\text{ million barrels per year} \\ \text{2018: }&11,200 + 12,200 + 12,300 = 35,700 \text{ (000 barrels per day)} \\ &35.7\text{m} \times 365 = 13,030.5\text{ million barrels per year}\end{aligned}

**Step 2:** Use the price of oil in both years to calculate the total value of oil generated in each year.

\text{1988: }\$40 \times 11,023\text{m} = \$440,920\text{ million}

\text{2018: }\$61.5 \times 13,030.5\text{m} = \$801,375.75\text{ million}

**Step 3:** Finally, calculate the percentage increase.

\dfrac{801,375.75 - 440,920}{440,920} \approx \bold{81.8\%\text{ increase}}

**Video Solutions**

**Question 6**

What was the difference in the value of oil generated per day by Saudi Arabia and Russia over the period?

** A: $84.01m**

B: $82.34m

C: $85.40m

D: $83.64m

**Written Solutions**

**Step 1;** First, find the difference in the output for each year. We need to be consistent with our subtraction, so we will do Saudi Arabia – Russia.

\begin{aligned}\text{1988: }& 10,100 - 9,700 = 400 \text{ (000’s of barrels per day)} \\ \text{1998: }& 10,600 - 11,500 = -900 \text{ (000’s of barrels per day)} \\ \text{2008: }& 11,300 - 11,000 = 300 \text{ (000’s of barrels per day)} \\ \text{2018: }& 12,200 - 11,200 = 1000 \text{ (000’s of barrels per day)}\end{aligned}

**Step 2:** Multiply each figure by the price per barrel, making sure to first convert from thousands of barrels to single barrels.

\begin{aligned}\text{1988: }&400,000 \times \$40 = \$16\text{m per day} \\ \text{1998: }&-900,000 \times \$19.5 = -\$17.55\text{m per day} \\ \text{2008: }&300,000 \times \$80.2 = \$24.06\text{m per day} \\ \text{2018: }&1,000,000 \times \$61.5 = \$61.5\text{m per day}\end{aligned}

**Step 3:** Add up these, making sure to include the sign in front of each difference.

\$16\text{m } - \$17.55\text{m } + \$24.06\text{m } +\$61.5\text{m } = \bold{\$84.01\text{m per day}}

**Video Solutions**

**Question 7**

What proportion of oil production in 2008 and 2018 can be attributed to Saudi Arabia?

** A: 33.5%**

B: 33.3%

C: 33.0%

D: 32.8%

**Written Solutions**

**Step 1: ** Calculate Saudi Arabia’s combined oil output in 2008 and 2018.

11,300 + 12,200 = 23,500\text{ (000 barrels per day)}

**Step 2:** Calculate the total output of the three countries in both years combined.

12,100 + 11,300 + 11,000 + 12,300 + 12,200 + 11,200 = 70,100\text{ (000 barrels per day)}

**Step 3:** Calculate the proportion. Proportion has no units, so provided we have continued to work in the same units (000’s of barrels per day) there is no need to multiply by 1000 or calculate the number of barrels per year.

\dfrac{23,500}{70,100} \times 100 \approx \bold{33.5\%}

**Video Solutions**

**Question 8**

Based on the percentage change in oil price and output from 2008 to 2018, what is the likely value of Russia’s total oil output in 2028?

A: $187bn

**B: $196bn**

C: $193bn

D: $180bn

**Written Solutions**

**Step 1:** Calculate the percentage decrease in oil price from 2008 to 2018 and apply it to the value in 2018 to get an estimate for 2028.

\dfrac{\$61.5 - \$80.2}{\$80.2} \times 100 = 23.3\%\text{ decrease}

\text{Decrease }\$61.5\text{ by }23.3\% \equiv \$61.5 \times 0.767 \approx \$47.17

**Step 2:** Calculate the percentage increase in oil output for Russia from 2008 and 2018, and apply it to the 2018 figure to get an estimate for 2028.

\dfrac{11,200 - 11,000}{11,000} \times 100 = 1.8\%\text{ increase}

\text{Increase }11,200\text{ by } 1.8\% \equiv 11,200 \times 1.018 \approx 11,402\text{ (000 barrels per day)}

**Step 3:** Calculate the prediction for 2028 based on these figures, making sure to convert our barrels figure to single barrels per year.

11,402 \times 1000 \times 365 \approx 4,162\text{m barrels per year}

4,162\text{m } \times \$47.17 \approx \bold{\$196\text{bn}}

**Video Solutions**

**Question 9**

From 1990 to 2000, Cod catches increased by 12%, Salmon by 25,000 metric tons and Haddock by 12,000. What is the percentage increase in the total catch of all three fishes from 1990 to 2000?

A: 14.1%

B: 14.3%

**C: 14.6%**

D: 14.8%

**Written Solutions**

**Step 1:** Calculate the new quantities of all three fish. A 12% increase for Cod is a multiplier of 1.12.

\begin{aligned}\text{Cod: }& 130,000 \times 1.12 = 145,600\text{ metric tons} \\ \text{Haddock: }&120,000 + 25,000 = 145,000\text{ metric tons} \\ \text{Salmon: }&110,000 + 12,000 = 122,000\text{ metric tons}\end{aligned}

**Step 2:** Add up the total catch in 1990 and 2000.

\text{1990: }130,000 + 120,000 + 110,000 = 360,000\text{ metric tons}

\text{2000: }145,600 + 145,000 + 122,000 = 412,600\text{ metric tons}

**Step 3: **Calculate the overall percentage increase.

\dfrac{412,600 - 360,000}{360,000} \times 100 = \bold{14.6\%}

**Video Solutions**

**Question 10**

Over 1970 and 1980 combined, what is the value of the Cod catch compared to the Salmon catch. Express this as a ratio.

A: 1: 1.40

**B: 1: 1.43**

C: 1: 1.38

D: 1: 1.35

**Written Solutions**

**Step 1: ** Calculate the value of the combined Cod catch. We need to convert from metric tons to stones before we can multiply by the price, this means we must multiply by 157.5.

150,000 + 140,000 = 290,000

290,000 \times 157.5 \times \pounds13.80 \approx \pounds630\text{m}

**Step 2: ** Calculate the value of the Salmon catch in the same way.

160,000 + 130,000 = 290,000

290,000 \times 157.5 \times \pounds19.70 \approx \pounds900\text{m}

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

\begin{aligned}\text{Cod} &: \text{Salmon} \\ \pounds630\text{m} &: \pounds900\text{m} \\ \dfrac{630}{630} &: \dfrac{900}{630} \\ 1 &: 1.43\end{aligned}

**Video Solutions**

**Question 11**

Pounds (weight) of fish are bought in the same quantities as the simplified ratio in which the three fish were caught in 1960. What is the total cost of this purchase? Give your answer to the nearest pound.

**A: £41**

B: £40

C: £39

D: £38

**Written Solutions**

**Step 1:** Write down and simplify the ratio of catches in 1960.

\begin{gathered}\text{Cod}:\text{Haddock}:\text{Salmon} \\ 120,000: 130,000: 140,000 \\ \text{Divide by 10,000} \\ 12: 13: 14 \end{gathered}

**Step 2:** So, we need to buy 12 pounds of Cod, 13 of Haddock and 14 of Salmon.

First, convert the prices into per pound rather than per stone, by dividing by 14.

\begin{aligned}\text{Cod:}&\pounds13.80 \div 14 = \pounds0.99 \\ \text{Haddock:}&\pounds10.40 \div 14 = \pounds0.74 \\ \text{Cod:}&\pounds19.70 \div 14 = \pounds1.41 \end{aligned}

Next, multiply each of these by the amount required, then find the total.

\begin{aligned}\text{Cod: }&\pounds0.99 \times 12 = \pounds11.88 \\ \text{Haddock: }&\pounds0.74 \times 13 = \pounds9.62 \\ \text{Salmon: }&\pounds1.41 \times 14 = \pounds19.74\end{aligned} \pounds11.88 + \pounds9.62 + \pounds19.74 \approx \bold{\pounds41}

**Video Solutions**

**Question 12**

What is the total value of the catch in 1960 and 1970 combined if costs of 12% and 17% must be deducted respectively?

A: £1,655m

B: £2,010m

C: £1,930m

**D: £1,740m**

**Written Solutions**

**Step 1:** Calculate the combined catch for each fish from 1960 to 1970, making sure to reduce by 12% (multiplier of 0.88) and 17% (multiplier of 0.83) respectively.

\begin{aligned}\text{Cod: }& (0.88 \times120,000) + (0.83 \times 150,000) = 230,100 \\ \text{Haddock: }&(0.88 \times 130,000) + (0.83 \times 190,000) = 272,100 \\ \text{Salmon: }&(0.88 \times 140,000) + (0.83 \times 160,000) = 256,000\end{aligned}

**Step 2:** Convert this to stones by multiplying by 157.5, and multiply by the price of one stone to calculate the cost.

\begin{aligned}\text{Cod: }&230,100 \times 157.5 \times 13.80 = \pounds500.1\text{m} \\ \text{Haddock: }&272,100 \times 157.5 \times 10.40 = \pounds445.7\text{m} \\ \text{Salmon: }&256,000 \times 157.5 \times 19.70 = \pounds794.3\text{m}\end{aligned}

**Step 3:** Add these up to get approximately **£1,740m**.

**Video Solutions**

**Question 13**

What is the yearly saving over 5 years if buying a petrol car and paying yearly compared to a diesel car paid monthly?

A: £19.20

B: £19.00

C: £18.80

**D: £18.60**

**Written Solutions**

**Step 1: **Calculate the cost of a petrol car paid over 5 years with yearly payments.

\begin{gathered}\text{first year} + \text{4 years of yearly payments} = \text{total cost} \\ \pounds143 + 4(\pounds128) = \pounds655 \end{gathered}

**Step 2:** Calculate the cost of a diesel car paid over 5 years with monthly payments.

\begin{gathered}\text{first year} + \text{4 years of monthly payments} = \text{total cost} \\ \pounds160 + (4 \times 12 \times \pounds12.25) = \pounds748 \end{gathered}

**Step 3:** Calculate the yearly saving. Find the difference and then divide by 5 years.

\pounds748-\pounds655=\pounds93

\pounds93 \div 5 = \bold{\pounds18.60}

**Video Solutions**

**Question 14**

Yearly payments increase by 12%. For which engine types are monthly payments now cheaper if paid for over 4 years?

**A: Petrol & Diesel**

B: Petrol & Alternative

C: Diesel & Alternative

D: All three

**Written Solutions**

**Step 1: **Calculate the new cost of yearly payments due to a 12% increase. This corresponds to a multiplier of 1.12.

\begin{aligned} \text{Petrol: } \pounds128 \times 1.12 &= \pounds143.36 \\ \text{Diesel: } \pounds135 \times 1.12 &= \pounds151.20 \\ \text{Alternative: } \pounds95 \times 1.12 &= \pounds106.40 \end{aligned}

**Step 2:** Calculate the total cost of payments over 4 years for each engine type, if paid yearly with the new yearly costs.

\begin{aligned} \text{Petrol: } \pounds143 + (3 \times \pounds143.36) &= \pounds573.08 \\ \text{Diesel: } \pounds160 +(3 \times \pounds151.20) &= \pounds613.60 \\ \text{Alternative: } \pounds113 + (3 \times \pounds106.40) &= \pounds432.20 \end{aligned}

**Step 3:** Calculate the total cost of payments over 4 years for each engine type if paid monthly.

\begin{aligned} \text{Petrol: } \pounds143 + (12 \times 3 \times \pounds11.50) &= \pounds557\\ \text{Diesel: } \pounds160 +(12 \times 3 \times \pounds12.25) &= \pounds601 \\ \text{Alternative: } \pounds113 + (12 \times 3 \times \pounds9.75) &= \pounds464 \end{aligned}

**Step 4:** Compare yearly and monthly totals to find that **Petrol and Diesel** are now cheaper.

**Video Solutions**

**Question 15**

Two new cars, one diesel and one alternative are valued at £50k each and have their tax paid yearly over 7 years. What is the difference between the proportion of the total amount paid due to valuation on the two cars?

A: 10.1%

B: 8.3%

**C: 8.7%**

D: 9.4%

**Written Solutions**

**Step 1:** Calculate the total paid due to valuation. This is the same for both cars and all five years must be paid as these cars are paid over 7 years.

\pounds 165 \times 5 = \pounds825

**Step 2:** Calculate the total payments over 7 years for both cars.

\begin{gathered}\text{initial payment}+(\text{6 years of future payments}) + \text{valuation} = \text{total} \\ \text{Diesel: }\pounds160+(6 \times \pounds135) + \pounds825 = \pounds1,795 \\ \text{Alternative: }\pounds113+(6 \times \pounds95) + \pounds825 = \pounds1,508\end{gathered}

**Step 3: **Calculate the proportion of the total paid due to valuation on each car.

\begin{aligned}\text{Diesel: }\dfrac{\text{Cost due to valuation}}{\text{Total cost}}=\dfrac{\pounds825}{\pounds1,795} &= 0.46 \text{ or }46\% \\ \text{Alternative: } \dfrac{\text{Cost due to valuation}}{\text{Total cost}} = \dfrac{\pounds825}{\pounds1,508} &= 0.547 \text{ or }54.7\%\end{aligned}

Finally find the difference between these two proportions to get 54.7\% - 46\% = \bold{8.7\%}

**Video Solutions**

**Question 16**

Business sales increased by 12% from 2015 to 2016 while personal sales increased by 21%. What was the ratio of total petrol and electric sales in 2015 compared to 2016?

A: 1: 1.21

B: 1: 1.24

C: 1: 1.14

**D: 1: 1.17**

**Written Solutions**

**Step 1: **We need to calculate the usage stats for 2015 petrol and electric engines. We know business and personal increased by 12% and 21% respectively. These are multipliers of 1.12 and 1.21, and to get back to 2015 figures from 2016 we must divide by the multiplier.

\begin{aligned}\text{Petrol: Business: }& 6,170 \div 1.12 = 5,509 \\ \text{Personal: }&7,425 \div 1.21 = 6,136 \\ \text{Electric: Business: }& 235 \div 1.12 = 210 \\ \text{Personal: }&170 \div 1.21 = 141 \end{aligned}

**Step 2:** Add up the total usage for electric and petrol for both 2015 and 2016.

\text{2015: }5,509+6,136+210+141 = 11,996

\text{2016: }6,170+7,425+235+170 = 14,000

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

\begin{aligned} 2015 &: 2016 \\ 11,996 &: 14,000 \\ \dfrac{11,996}{11,996} &: \dfrac{14,000}{11,996} \text{ (divide both parts by the lower number)} \\ 1 &: 1.17 \end{aligned}

**Video Solutions**

**Question 17**

What is the percentage increase in the number of bags produced by Vietnam and Indonesia combined from 2007 to 2017?

A: 99%

B: 95%

**C: 93%**

D: 91%

**Written Solutions**

**Step 1:** Calculate for 2007.

22\% + 7\% = 29\%\text{ of }75\text{m bags} = 0.29 \times 75\text{m} = 21.75\text{m}

**Step 2: **Make the same calculation for 2017.

29\%+11\% = 40\%\text{ of }105\text{m bags} = 0.40 \times 105\text{m} = 42\text{m}

**Step 3:** Calculate the percentage increase.

\dfrac{42-21.75}{21.75} \times 100 = \bold{93\%}

**Video Solutions**

**Question 18**

In 2017, 2m bags of Colombian coffee and 3m bags of Indonesian have to be discarded. What is the percentage change in the combined output of Colombia and Indonesia from 2007 to 2017 in this case?

**A: 7.7% increase**

B: 7.2% increase

C: 5.4% decrease

D: 10.2% increase

**Written Solutions**

**Step 1:** Calculate the total output in 2007.

18\% + 7\% = 25\%

25\%\text{ of }75\text{m} \equiv 0.25 \times 75\text{m} = 18.75\text{m}

**Step 2:** Calculate the total output in 2017, making sure to deduct 5m bags in total.

13\% + 11\% = 24\%

24\%\text{ of }105\text{m} \equiv 0.24 \times 105\text{m} = 25.2\text{m}

25.2\text{m} - 5\text{m} = 20.2\text{m}

**Step 3:** Find the percentage change between the two periods.

\dfrac{20.2 - 18.75}{18.75} \times 100 = \bold{7.7\%\text{ increase}}

**Video Solutions**

**Question 19**

What was the difference between the highest and second highest increases in the output of a country over the period?

A: 4.95m

**B: 4.35m**

C: 4.60m

D: 4.05m

**Written Solutions**

**Step 1:** Identify candidates. The percentage share for Colombia decreases, and which this still may be an increase in output it will not be one of the highest. We should calculate the increase in output for the other three countries however.

\begin{aligned}\text{Brazil - 2007: }&53\%\text{ of }75\text{m} = 39.75\text{m} \\ \text{Brazil - 2017: }&47\%\text{ of }105\text{m} = 49.35\text{m} \\ \text{Difference: }&49.35 - 39.75 = 9.6\text{m}\end{aligned}

\begin{aligned}\text{Vietnam - 2007: }&22\%\text{ of }75\text{m} = 16.5\text{m} \\ \text{Vietnam - 2017: }&29\%\text{ of }105\text{m} = 30.45\text{m} \\ \text{Difference: }&30.45 - 16.5 = 13.95\text{m}\end{aligned}

\begin{aligned}\text{Indonesia - 2007: }&7\%\text{ of }75\text{m} = 5.25\text{m} \\ \text{Indonesia - 2017: }&11\%\text{ of }105\text{m} = 11.55\text{m} \\ \text{Difference: }&11.55 - 5.25 = 6.3\text{m}\end{aligned}

**Step 2:** So the difference between the highest two increases is 13.95 - 9.6 = \bold{4.35}

**Video Solutions**

**Question 20**

A single bag of coffee weighs 60kg. 1 kg sells for £5.70. What is the difference in value of Brazil’s harvest from 2007 to 2017.

A: £3,314.3m

B: £3,019.5m

C: £3,154.8m

** D: £3,283.2m**

**Written Solutions**

**Step 1:** First calculate the quantity produced by Brazil in both years.

\begin{aligned}\text{2007: }&53\%\text{ of }75\text{m} = 39.75\text{m} \\ \text{2017: }&47\%\text{ of }105\text{m} = 49.35\text{m}\end{aligned}

**Step 2:** Find the difference between 2017 and 2007

49.35 - 39.75 = 9.6\text{m}

**Step 3:** Finally work out the value of this by multiplying by 60 (to convert to kilograms) and then by the price per kilogram.

9.6\text{m} \times 60 \times\pounds5.70 = \bold{\pounds3,283.2{m}}