## Worked Solutions – Test 2

**Question 1**

The “Other” category is comprised of three subsectors that share funding in the ratio 2: 2: 6. What is the ratio of spending in 2006 compared to 2016 of the third subsector?

**A: 1: 1.44**

B: 1: 1.23

C: 1: 1.36

D: 1: 1.19

**Written Solutions**

**Step 1: **Simplify the ratio if possible and work out the total number of shares.

\begin{aligned}2&: 2: 6 \\ 1&: 1: 3 \text{(divide by 2)} \end{aligned}

There is a total of 5 shares. Looking at 2016, this segment accounts for 35% of the total.

35 \div 5 = 7

So, we multiply the ratio by 7 to get the real percentage share of each subsector in “Other”

\begin{aligned}1&: 1: 3 \\ 7\%&: 7\%: 21\% \end{aligned}

So, the third subsector accounts for 21% of the total spending in 2016.

**Step 2:** Complete the same process for 2006.

\begin{aligned}1&: 1: 3 \\ 30 \div 5& = 6 \text{(so multiply by 6)} \\ 6\%&: 6\%: 18\%\end{aligned}

So, the third subsector accounts for 18% of the total spending in 2006.

**Step 3:** Form the ratio of spending in 2006 compared to 2016 for the third subsector.

\begin{aligned} \text{2006 spending} &: \text{2016 spending} \\ 18\% \text{ of } \pounds491.8bn &: 21\% \text{ of } \pounds606.6bn \\ 0.18 \times 491.8 &: 0.21 \times 606.6 \\ 88.524 &: 127.386 \\ 1 &: 1.44 \text{ (divide through by 88.524)}\end{aligned}

**Video Solutions**

**Question 2**

What was the highest increase in value of any sector over the period?

A: £54.3bn

B: £55.9bn

C: £62.1bn

**D: £69.2bn**

**Written Solutions**

**Step 1:** identify which sectors increase in value over the period.

Clearly Pensions, Health Care and Other increase in value over the period. The others may still increase in value even though their percentage share decreases, so we can do a quick calculation to check.

\begin{aligned}\text{Welfare: }2006:& 0.16 \times 491.8 = 78.688 \\ 2016:& 0.09 \times 606.6 = 54.594\end{aligned}

\begin{aligned}\text{Education: }2006:& 0.13 \times 491.8 = 63.934 \\ 2016:& 0.07 \times 606.6 = 42.462\end{aligned}

**Step 2:** Calculate the change in value for all three sectors that increase.

First calculate this for the Pensions sector:

\begin{gathered} \text{value in 2016} - \text{value in 2006} = \text{increase} \\ 26\% \text{ of }606.6 - 18\% \text{ of }491.8 = \text{increase} \\ (0.26 \times 606.6) - (0.18 \times 491.8) = 157.716 - 88.524 = \pounds69.192bn \text{ increase} \end{gathered}

Similarly, for the other two sectors:

\begin{aligned}\text{Health Care: }&0.23 \times 606.6 - 0.17 \times 491.8 = 139.518 - 83.606 &= 55.912 \\ \text{Other: }&0.35 \times 606.6 - 0.30 \times 491.8 = 212.32 - 147.54 &= 64.78 \end{aligned}

We conclude that Pensions had the largest increase of £69.192bn.

**Video Solutions**

**Question 3**

Assuming an inflation rate of 2% per year, what was the change in welfare spending over the period in real terms as a percentage of spending in 2006?

**A: 52.5% decrease**

B: 62.4% decrease

C: 12.4% increase

D: 49.3% decrease

**Written Solutions**

**Step 1: **Calculate the value of total spending in 2006 under an inflation rate of 2%. An increase of 2% corresponds to a multiplier of 1.02.

\text{2006 spending} \times 1.02^{10} = \text{value of 2006 spending in 2016 accounting for inflation}

1.02^{10} is shorthand for using the multiplier of 1.02 ten times, once for each year.

So inflation adjusted value of 2006 spending is:

491.8 \times 1.02^{10} = 599.5

**Step 2: **Calculate the total spending on welfare for 2006 and 2016 based on the new figures.

\text{2006: }16\% \text{ of } 599.5 = 0.16 \times 599.5 = 95.92

\text{2016: }9\% \text{ of } 606.6 = 0.09 \times 606.6 = 54.59

A fall of £41.33bn from 2006.

**Step 3: **Calculate this change as a percentage of the actual 2006 spending on welfare.

\dfrac{ \text{Real terms change}}{\text{Welfare spending in 2006}}\times 100 = \dfrac{-41.33}{0.16 \times 491.8} \times 100 = \bold{52.5\% \text{ decrease}}

**Video Solutions**

**Question 4**

If spending on education in **2011** increased by 14.2bn from 2006, and spending on Welfare, Other and Health Care combined rose by 5% from **2011** to 2016, what percentage of spending in 2011 was allocated to Pensions?

A: 19.6%

B: 18.7%

**C: 20.6%**

D: 21.1%

**Written Solutions**

**Step 1: **Calculate the spend on education in 2011. Start by adding the increased spending on to the 2006 figures.

13\% \text{ of }491.8 = 0.13 \times 491.8 = 63.934

63.934 + 14.2 = 78.134 \text{ (education spending in 2011)}

Next, we need to work out the percentage increase to work out the share of total spending education takes up in 2011

\dfrac{78.134 - 65.1}{65.1}\times 100 = 20.02\%\text{ increase}

Now apply this percentage increase to the share of spending in 2006. A 20.02% increase is a multiplier of 1.2002.

0.13 \times 1.2002 \times 100 = 15.6\%

**Step 2:** Since all three of welfare, other and health care increased by 5% from 2011 to 2016 we can add them up and do the calculation.

9\% + 35\% + 23\% = 67\%

These three increased by 5% from 2011 to 2016. That was a multiplier of 1.05. To return to the 2011 figures we must divide by 1.05.

67\% \div 1.05 = 63.8\%

**Step 3: ** Finally, we can add these up and subtract to calculate the remaining share which must be allocated to pensions.

100\% - 63.8\% - 15.6\% = \bold{20.6\%}

**Video Solutions**

**Question 5**

What is the ratio of costs of Gold if bought as follows: 1kg in July 2015, 1 troy ounce (to) in January 2016 and 1 pennyweight (pwt) in July 2016. 20 pennyweights are equivalent to 1 troy ounce.

A: 643: 32.15: 1

**B: 479.4: 14.27: 1**

C: 534.6: 18.45: 1

D: 342.7: 20.42: 1

**Written Solutions**

**Step 1:** Form ratios for the cost of 1kg of gold in each of the time periods, and for the conversion of units that are required.

\begin{gathered}\pounds24,026: \pounds22,979: \pounds32,193 \\ 1\text{kg}: 32.15\text{to}: 32.15 \times 20 \text{pwt}\end{gathered}

The ratio of units is all equal to 1kg each as per the conversions given.

**Step 2: **We now divide each part of the ratio of costs in kg by the unit conversions to get the cost per unit of measurement.

\begin{gathered}\dfrac{\pounds24,026}{1kg}: \dfrac{\pounds22,979}{32.15\text{to}}: \dfrac{\pounds32,193}{643\text{pwt}} \\ 24,026: 714.74: 50.07 \end{gathered}

**Step 3:** The final step is the simplify the ratio by dividing through by the smallest value, in this case 50.07, which gives us

\bold{479.4 : 14.27 : 1}

**Video Solutions**

**Question 6**

In Jan 2018 I spend as much of $10,000 as possible on whole troy ounces of platinum and palladium in the ratio 1: 1. What is my total spend as a proportion of the combined cost per kg of both metals?

A: 10.3%

**B: 12.4%**

C: 11.7%

D: 9.5%

**Written Solutions**

**Step 1: **Calculate the combined cost per troy ounce of platinum and palladium.

First, we need the combined cost in kilograms:

\pounds24,743 + \pounds 17,591 = \pounds42,334 \text{ (per kg)}

Divide through by the conversion of 1kg = 32.15 troy ounces to get the combined cost per troy ounce

\pounds42,334 \div 32.15 = \pounds1,317 \text{ (per troy ounce)}

**Step 2:** Convert the $10,000 to pounds and work out the total spend.

Using the conversion £1 = $1.62:

\$10,000 \equiv \pounds(10,000 \div 1.62) = \pounds6,173

We want whole troy ounces so discard the decimal place on the following calculation:

\begin{gathered}\text{Total available cash} \div \text{combined price per troy ounce} \\ \pounds6,173 \div \pounds1,317 = 4.68 \end{gathered}

Finally multiply the cost by 4 to calculate the total spend:

\pounds1,317 \times 4 = \pounds5,268

**Step 3:** Divide the total spend by the combined cost per kg of both metals to calculate the proportion.

\dfrac{\pounds5,268}{\pounds42,334}\times 100 = \bold{12.4\%}

**Video Solutions**

**Question 7**

The dollar follows the value of palladium precisely. What is the price of 1kg of each metal in dollars when the pound is weakest against the dollar to the nearest $10?

A: $53,240

B: $54,970

C: $45,820

**D: $60,850**

**Written Solutions**

**Step 1:** The pound is weakest when the value of palladium is lowest. That would be in January 2016. So, calculate the percentage change in price of palladium from Jan 2016 to Jan 2018.

\dfrac{\pounds17,591 - \pounds12,121}{\pounds12,121} \times 100 = 45.1\%

**Step 2: **To increase from 2016 to 2018, this was a multiplier of 1.451. Since the value of the dollar follows the price of palladium we can use this multiplier to find the value of the dollar in 2016. So, we divide by 1.451 to return from the 2018 value to the 2016 value.

** ** \$1.62 \div 1.451 = \$1.12

** ****Step 3: **Add up the price of all three metals in Jan 2016 and convert to dollars.

** **\pounds22,979 + \pounds19,231 + \pounds 12,121 = \pounds54,331

\pounds54,331 \times 1.12 = \bold{\$60,850}

**Video Solutions**

**Question 8**

If the rate of change in prices from July 2017 to January 2018 continues, what will the total value of 1kg of each metal be in July 2018 in dollars to the nearest thousand, if there is no change in the exchange rate?

**A: $119k**

B: $113k

C: $116k

D: $114k

**Written Solutions**

**Step 1:** Calculate the percentage change from July 2017 to January 2018 for all three metals.

\begin{aligned}\text{Gold: }&\dfrac{\pounds31,128- \pounds30,788}{\pounds30,788} \times 100 = 1.1\% \text{ increase} \\ \text{Platinum: }&\dfrac{\pounds24,743- \pounds22,794}{\pounds22,794} \times 100 = 8.6\% \text{ increase} \\ \text{Palladium: }&\dfrac{\pounds17,591- \pounds20,788}{\pounds20,788} \times 100 = 15.4\% \text{ decrease}\end{aligned}

**Step 2: **Apply this percentage change from January 2018 to July 2018.

\begin{aligned}\text{Gold: }&1.1\% \text{ increase} = \text{ multiplier of } 1.011 \\ &1.011 \times \pounds31,128 = \pounds31,470 \\ \text{Platinum: }&8.6\% \text{ increase} = \text{ multiplier of } 1.086 \\ &1.086 \times \pounds24,743 = \pounds26,871 \\ \text{Palladium: }&15.4\% \text{ decrease} = \text{ multiplier of } 0.846 \\ &0.846 \times \pounds17,594 = \pounds14,885 \end{aligned}

**Step 3:** Add up the values for July 2018 and convert into dollars.

\pounds31,470 + \pounds26,871 +\pounds14,885 = \pounds73,229

\pounds73,229 \times 1.62 = \bold{\$119k}

**Video Solutions**

**Question 9**

Trains to Birmingham are delayed by 24% of their duration and there is an additional wait of 25 minutes before the train leaves for Derby. £10 is refunded for every full 15 minutes the train was delayed. What is the refund as a percentage of standard ticket costs on the usual fastest train from Bath to York?

A: 25.6%

**B: 22.8%**

C: 20.8%

D: 29.3%

**Written Solutions**

**Step 1:** Identify the fastest train from Bath to York.

10:45 - 05:58 = 04:47

11:08 - 06:45 = 04:23

11:57 - 07:35 = 04:22 \checkmark

**Step 2:** Calculate the total duration of delays on this train.

The duration from Bath to Birmingham on the 07:35 train is:

09:00 - 07:35 = 01:25

Now calculate 24% of that duration to work out the delay

24\%\text{ of }01:25 = 0.24 \times 85\text{ minutes } = 20.4 \text{ minutes}

This means there is a total delay of 20.4 + 25 = 45.4 \text{ minutes}

**Step 3:** Calculate the refund. 45.4 minutes means a total of three 15 minutes blocks which are refunded at £10 each.

Now calculate that as a percentage of the standard ticket cost for this train:

\dfrac{\pounds30}{\pounds131.70} \times 100 = \bold{22.8\%}

**Video Solutions**

**Question 10**

What is the difference between the lowest and highest costs of travel per hour on any train running from Birmingham to York?

A: £23.46

**B: £26.77**

C: £31.70

D: £35.43

**Written Solutions**

**Step 1:** Calculate the duration of all trains from Birmingham to York and identify the fastest and slowest trains.

10:45 - 07:20 = 03:25 \checkmark

11:08 - 08:10 = 02:58

11:57 - 09:00 = 02:57 \checkmark

12:13 - 09:10 = 03:03

**Step 2:** The highest cost ticket is £137.10 by first class, and the lowest is £67.30 standard class. Use these with the slowest and fastest trains respectively to calculate the costs per hour.

\text{Cost per hour} = \dfrac{\text{Cost of a ticket}}{\text{Duration of train}}

To not cause confusion between 03:30 and 3.5 hours we first convert times into minutes. After the calculate we will convert back into hours by multiplying by 60.

\begin{aligned}\text{Lowest cost: }&03:25 = 180 + 25 = 205\text{ minutes} \\ &\dfrac{\pounds67.30}{205}\times 60 = \pounds19.70 \\ \text{Highest cost: }&02:57 = 120 + 57 = 177\text{ minutes} \\ &\dfrac{\pounds137.10}{177}\times 60 = \pounds46.47\end{aligned}

**Step 3: ** Finally calculate the difference: \pounds46.47 - \pounds19.70 = \bold{\pounds26.77}

**Video Solutions**

**Question 11**

Trains on all services exiting Bath are delayed by 5 minutes on arrival to each subsequent stop. A businessman travels from Bath to York and needs to spend an hour in Birmingham for a meeting. He wants to get to York as early as possible – what percentage of his journey time is due to delays and his stop in Birmingham?

A: 15%

**B: 20%**

C: 25%

D: 10%

**Written Solutions**

**Step 1:** We need to calculate the new arrival times in Birmingham. Each train leaving Bath is delayed by 5 minutes for three stops so that is a 15-minute delay.

The new times are:

05:58 - 07:20 \rightarrow 07:35

06:45 - 08:10 \rightarrow 08:25

06:55 - 08:20 \rightarrow 08:35

07:35 - 09:00 \rightarrow 09:15

**Step 2:** We now need to account for his hour meeting – which services can he get that give him enough time to have this meeting?

Only the first service fits this requirement, with an hour meeting he would finish at 07:35 + 1:00 = 8:35 with two trains to choose from the leave Birmingham.

**Step 3:** Decide which of the 07:35 from Bath or 09:10 from Birmingham arrives to York earliest.

The 07:35 is delayed by 5 minutes for each stop which is a 35-minute delay. It therefore arrives at 12:32.

The 09:10 from Birmingham is not delayed as it did not depart Bath, and arrives at 12:13, making this the faster train.

**Step 4:** Calculate his total journey time, duration of delays and stops, and finally the percentage required.

\begin{aligned}\text{Journey time: }&12\colon13 - 05\colon58 = 06\colon15 \equiv 375\text{ minutes} \\ \text{Duration of delays and stops: }&(5\text{minutes} \times 3) + 1 \text{ hour} = 75 \text{ minutes}\end{aligned}

\dfrac{\text{Duration of delays and stops}}{\text{Total journey time}} \times 100 = \dfrac{75\text{ minutes}}{375\text{ minutes}} \times 100 = \bold{20\%}

**Video Solutions**

**Question 12**

What is the distance from Bath to York given that the 07:35 train travels at 67mph to Birmingham, then 41mph to Derby and 53mph thereafter?

A: 232.4 miles

B: 237.2 miles

**C: 241.3 miles**

D: 239.8 miles

**Written Solutions**

**Step 1:** Calculate the duration of each stage of the 07:35 train.

\begin{aligned}\text{Bath to Birmingham: }& 09\colon00 - 07\colon35 = 01\colon25 \equiv 85\text {minutes} \\ \text{Birmingham to Derby: }& 09\colon50 - 09\colon00 = 50\text{ minutes} \\ \text{Derby to York: }& 11\colon57 - 09\colon50 = 02\colon07 \equiv 127\text{ minutes} \end{aligned}

**Step 2:** Calculate the distance travelled in each stage using the formula for speed, distance and time.

Distance = Speed \times Time

After multiplying speed by time in minutes, we must divide by 60 to turn it back into hours.

\text{Stage 1: }(85\text{ minutes} \times 67\text{mph}) \div 60\text{ minutes}= 94.9\text{miles}

\text{Stage 2: }(50\text{ minutes} \times 41\text{mph})\div 60\text{ minutes}= 34.2\text{miles}

\text{Stage 3: }(127\text{ minutes} \times 53\text{mph}) \div 60\text{ minutes}= 112.2\text{miles}

So the total distance is 94.9 + 34.2 + 112.2 = \bold{241.3\text{ miles}}

**Video Solutions**

**Question 13**

What is the best average price per month for a contract with at least 5 times as many texts as minutes?

**A: £35.42**

B: £36.17

C: £46.21

D: £43.78

**Written Solutions**

**Step 1: **Compare the texts and minutes for each contract to determine whether they are suitable. We multiply the number of minutes by 5 and look for the contracts where this is greater than the number of texts:

\begin{array}{c|c} \text{Red}& 1800\text{ minutes} \times 5 = 9000 > 8500 \checkmark \\ \text{Phonestore} & 900\text{ minutes} \times 5 = 4500 > 3000 \checkmark \\ \text{Mob4u} & 920\text{ minutes} \times 5 = 4600 \ngtr 4700 \\ \text{Eldron} & 1300\text { minutes} \times 5 = 6500 > 6200 \checkmark \\ \text{HeaHea} & 540\text{ minutes} \times 5 = 2700 \ngtr 2900\end{array}

**Step 2:** Calculate the average price per month for each of these contracts. First we should work out the total price of the contract over their duration.

\begin{array}{c|c}\text{Red} & \pounds220 + (24\text{months} \times \pounds26.25) = \pounds850 \\ \text{Phonestore} & \pounds251 + (16\text{ months} \times \pounds25.00) = \pounds651 \\ \text{Eldron} & \pounds245 + (9\text{ months} \times 18.99) = \pounds415.91\end{array}

**Step 3:** Finally divide each value by the number of months in the contract to get the average, and select our answer.

\begin{array}{c|c}\text{Red}& \pounds850 \div 24\text{ months} = \bold{\pounds35.42} \\ \text{Phonestore}& \pounds651 \div 18\text{ months} = \pounds36.17 \\ \text{Eldron}& \pounds415.91 \div 9\text{ months} = \pounds46.21 \end{array}

**Video Solutions**

**Question 14**

Monthly rates are due to rise by 4% a year for the next two years due to inflation. What will be the percentage increase in price of a 24-month contract with Red?

A: 4%

B: 5%

C: 8%

**D: 6%**

**Written Solutions**

**Step 1: **Calculate the total cost of a 24 month contract for Red.

\pounds220 + (24\text{ months} \times \pounds26.25) = \pounds850

**Step 2:** Calculate the new monthly cost due to two increases of 4%.

Increasing a value by 4% is the same as a multiplier of 1.04. If we need to do this over multiple years, in this case two, we multiply by 1.04^2

\pounds26.25 \times 1.04^2 = \pounds28.39

**Step 3:** Calculate the new cost of a 24 month contract under the new monthly rate.

\pounds220 + (24\text{ months} \times \pounds28.39) = \pounds901.36

**Step 4:** Calculate the percentage increase in price due to two years of inflation.

\dfrac{\pounds901.36 - \pounds850.00}{\pounds850.00}\times 100 = \bold{6\%}

**Video Solutions**

**Question 15**

Three friends use their unlimited data package in the ratio 5: 7: 11. Over 2 months the first friend used 15Gb and 20Gb respectively. What is the average expenditure on extra data per person per month?

A: £45.11

**B: £38.17**

C: £31.23

D: £34.74

**Written Solutions**

**Step 1:** The first friend uses 15Gb of data and their share of the ratio is 5. So, we multiply the ratio by 3 to get the other friends data usage. Similarly, we do this for the second month, but multiply by 4.

\begin{aligned}\text{Month 1: }&5:7:11 \\ 15:21:33 (\times3) \\ \text{Month 2: }&5:7:11 \\ &20:28:44 (\times4)\end{aligned}

From these ratios we can calculate the data usage above the allowances of the contract. We subtract 16Gb (the allowance mentioned) from each part of each ratio.

\text{Month 1: }0:5:17

\text{Month 2: }4:12:28

**Step 2:** Now sum up all additional data used over the period and calculate the average per person per month. Additional data is charged at £3.47 per Gb.

5+17+4+12+28 = 66Gb

66Gb \times \pounds3.47 = \pounds229.02

Now divide by 3\text{ people} \times 2\text{ months} = 6 to get the average per person per month.

\dfrac{\pounds229.02}{6} = \bold{\pounds38.17}

**Video Solutions**

**Question 16**

For a contract from Eldron, what is the percentage increase in the total cost of a contract if the user uses more than 1300 minutes for the first time in the 3^{rd} month, and then again in the 5^{th} month?

A: 7.3%

B: 9.4%

**C: 8.2%**

D: 6.1%

**Written Solutions**

**Step 1:** Calculate the usual total cost of a contract from Eldron.

\pounds245 + (9\text{ months} \times \pounds18.99) = \pounds415.91

**Step 2: **Now calculate the cost if it is first increased __after__ the 3^{rd} month to £22.39, and again after the 5^{th} month to £25.79. We will need to do this in three parts.

\begin{aligned}\text{First three months: }&\pounds245+(3\text{ months} \times \pounds18.99) = \pounds301.97 \\ \text{Next 2 months: }&2 \times \pounds22.39 = \pounds44.78 \\ \text{Last 4 months: }& 4 \times \pounds25.79 = \pounds103.16 \end{aligned}

**Step 3: **Finally add these up to find the new cost of the contract and calculate the percentage increase.

\pounds301.97 + \pounds44.78 + \pounds103.16 = \pounds449.91

\dfrac{\pounds449.91 - \pounds415.91}{\pounds415.91} \times 100 = \bold{8.2\%}

**Video Solutions**

**Question 17**

From 2016 – 2017 what was the approximate percentage change in profit for gas?

A: 3.2% increase

**B: 2.8% increase**

C: 2.7% decrease

D: 3.1% increase

**Written Solutions**

**Step 1:** Identify the revenue of gas in 2016 and 2017 from the graph.

2016 = \pounds475,000 2017 = \pounds490,000

**Step 2:** Calculate outgoings for each of these years.

Subtract £150,000 from revenue2016: \pounds475,000-\pounds150,000=\pounds325,000 2017: \pounds490,000-\pounds150,000=\pounds340,000

Calculate how many times £1000 goes into this value2016: \pounds325,000 \div \pounds1000 = 325 2017: \pounds340,000 \div \pounds1000 = 340

£290 is owed in outgoings for each £1000 so:

2016: 325\times\pounds290 = \pounds94,250 2017: 340\times\pounds290 = \pounds98,600

**Step 3: **Subtract the outgoings from revenue to get profit.

2016: \pounds475,000 - \pounds94,250 = \pounds380,750
2017: \pounds490,000 - \pounds98,600 = \pounds391,400
**Step 4:** Calculate the percentage change in profit.

\dfrac{\pounds391,400- \pounds380,750}{\pounds380,750}\times 100 = 2.8\% increase.

**Video Solutions**

**Question 18**

In 2018, what was the approximate ratio of profits of gas, water, and electric?

**A: 2: 1: 1.36**

B: 1.84: 1: 1.45

C: 2.2: 1: 1.32

D: 2.14: 1: 1.4

**Written Solutions**

**Step 1: **Identify the revenue of the three utilities in 2018 from the graph.

Gas = \pounds445,000 Water = \pounds420,000 Electric = \pounds475,000

**Step 2:** Calculate outgoings for each utility.

**Gas: **

Subtract 150,000 from the revenue\pounds445,000 - \pounds150,000 = \pounds295,000
Calculate how many times £1000 goes into this value\pounds295,000 \div \pounds1000 = 295
£290 is owed in outgoings for each £1000 so295 \times \pounds290 = \pounds85,550

**Water:**

Multiply £20,000 by 12 months in a year\pounds20,000 \times 12 = \pounds240,000

**Electric:**

By 2018 the initial outgoings of £200,000 have increased by 15% once. So we use the multiplier 1.15\pounds200,000 \times 1.15 = \pounds230,000

**Step 3:** Subtract the outgoings from revenue to get profit.

Gas = \pounds445,000 - \pounds85,550 = \pounds359,450 Water = \pounds420,000 - \pounds240,000 = \pounds180,000 Electric = \pounds475,000 - \pounds230,000 = \pounds245,000

**Step 4: **Form the ratio as follows.

Gas: Water: Electric \pounds359,450: \pounds180,000: \pounds245,000

Simplify the ratio by selecting water (as it has the lowest value) and dividing through.

\dfrac{\pounds359,450}{\pounds180,000}:\dfrac{\pounds180,000}{\pounds180,000}:\dfrac{\pounds245,000}{\pounds180,000}

Finally evaluate each fraction to obtain the simplified ratio

2: 1: 1.36

**Video Solutions**

**Question 19**

From 2020 revenues, gas is predicted to decrease by 20% in 2021, while water and electric will decrease by 15%. What would the approximate total profit of the utilities be in 2021?

A: £563k

B: £535k

**C: £518k**

D: £548k

**Written Solutions**

**Step 1:** Identify the revenue of the three utilities in 2020 from the graph.

Gas = \pounds490,000 Water = \pounds425,000 Electric = \pounds500,000

**Step 2:** Calculate revenues for 2021 using the multiplies 0.8 (20% reduction) and 0.85 (15% reduction) respectively.

Gas = \pounds490,000 \times 0.8 = \pounds392,000

Water = \pounds425,000 \times 0.85 = \pounds361,250 Electric = \pounds500,000 \times 0.85 = \pounds425,000

**Note: **Often in this situation we would have calculated water and electric as a combined value since they use the same multiplier. However, we have specific outgoings to deduct later so it is easier to keep them separate for now.

**Step 3:** Calculate the outgoings for 2021.

**Gas: **

Subtract 150,000 from the revenue

\pounds392,000 - \pounds150,000 = \pounds242,000

Calculate how many times £1000 goes into this value

\pounds242,000 \div \pounds1000 = 242

£290 is owed in outgoings for each £1000 so

242 \times \pounds290 = \pounds70,180

**Water:**

Multiply £20,000 by 12 months in a year

\pounds20,000 \times 12 = \pounds240,000

**Electric:**

By 2021 the initial outgoings of £200,000 have increased by 15% four times. So, we use the multiplier 1.15 four times, in other words:

\pounds200,000 \times 1.15^4 = \pounds349,800

**Note:** 1.15^4 is shorthand for 1.15 \times 1.15 \times 1.15 \times 1.15

**Step 4: **Subtract the outgoings from revenue to calculate profit.

Gas = \pounds392,000 - \pounds70,180 = \pounds321,820

Water = \pounds361,250 - \pounds240,000 = \pounds121,250 Electric = \pounds425,000 - \pounds349,800 = \pounds75,120

Finally, add up the individual profits to find the total:

\pounds321,820 + \pounds121,250 + \pounds75,120 = \pounds518,190

Which is £518k to the nearest thousand.

**Video Solutions**

**Question 20**

In 2019, approximately what was the proportion of total costs compared to revenue?

A: 39.8%

B: 33.3%

**C: 42.9%**

D: 40.4%

**Written Solutions**

**Step 1:** Identify the revenue of the three utilities in 2019 from the graph.

Gas = \pounds465,000 Water = \pounds445,000 Electric = \pounds480,000

**Step 2:** Calculate the outgoings for 2019.

**Gas: **

Subtract 150,000 from the revenue

\pounds465,000 - \pounds150,000 = \pounds315,000

Calculate how many times £1000 goes into this value

\pounds315,000 \div \pounds1000 = 315

£290 is owed in outgoings for each £1000 so

315 \times \pounds290 = \pounds91,350

**Water:**

Multiply £20,000 by 12 months in a year

\pounds20,000 \times 12 = \pounds240,000

**Electric:**

By 2019 the initial outgoings of £200,000 have increased by 15% twice. So, we use the multiplier 1.15 two times, in other words:

\pounds200,000 \times 1.15^2 = \pounds264,500

**Step 3:** Calculate the total costs and total revenue.

\text{Total costs }= \pounds91,350 + \pounds240,000 + \pounds264,500 = \pounds595,850

\text{Total revenue }= \pounds465,000 + \pounds445,000 + \pounds480,000 = \pounds 1,390,000

**Step 4: **Calculate the proportion of costs to revenue.

\dfrac{\text{Total costs}}{\text{Total revenue}}\times 100 = \dfrac{\pounds595,850}{\pounds1,390,000} \times 100 = \approx 42.9\%