QR-5 Percentages & Proportion – Homework

Explanation

Answer: D

Explanation: 

1. Convert the units into metres and seconds and find the average speed.

1 hour = 60 minutes = 60 x 60 seconds = 3600 s

10 km = 10,000 m

So in metres per second:

10000 / 3600 = 2.777… = 2.8 m/s

Common trap: Notice that the answers are all in m/s. He has run at 10 km/h so make sure that you don’t slip up and pick answer E without thinking.

Explanation

Answer: C

Explanation: 

1. Work out his old cost and his new cost and find the percentage difference

His old costs were the meal deal and coffee:

2.75 + 3.00 = £5.75

His new costs are:

1.50

Use the formula:

Multiplier = New Value / Old Value
Multiplier = 1.50 / 5.75 = 0.26….

0.26… = 74% percentage decrease – Option C.

Timing Tip: Percentage decrease is one of the most commonly tested skills in UCAT Quantitative Reasoning. Get to grips with the multiplier method to save precious time.

Timing Tip: Note that the percentage difference will be same for one day as for 5 – there is no need to work out the values for 5 days’ worth of purchases.

Explanation

Answer: C

Explanation: 

1. Find the speed at which he is typing and the time he is typing for.

He is working at 67.5% of his maximum of 80 words per minute.

67.5% of 80 = 54 words per minute.

He is typing for 4 and a half hours apart from a half-an-hour zoom call and 10 minutes total in breaks.

This means that, in minutes, he was working for 270 – 30 – 10 = 230 minutes.

This means he typed:

230 x 54 = 12,420 words in his pre-lunch session.

Common Trap: Note that he does not type at his maximum speed in this work session – if he did then the answer would be E.

Explanation

Answer: D

Explanation: 

1. Find the eco-shower’s water use and thus the other showers’ use.

The eco shower uses 25 ml per second.

If Harry showers for half the time between arriving home and starting work, he showers for 15 minutes.

15 x 60 x 25 = 22,500 ml.

This is 20% less than other models:

Use the formula:

New Value / Multiplier = Old Value

The multiplier for a 20% decrease is 0.8:

22500 / 0.8 = Old Value

28125ml = Old Value

In litres:

28125 / 1000 = 28.125L – answer D.

Common Trap: Remember when working out a percentage decrease to divide by the percentage decrease as a multiplier rather than by multiplying by the same percentage as a percentage increase. Using a multiplier of 1.2 would give 27L – lower than the true original value.

Explanation

Answer: D

Explanation: 

1. Find the average weekly sales in Year 1

123,545 in 5 weeks
123,545/5 = 24709 a week

2. Find the average weekly sales in Year 2 and thus the total:

New Value = Old Value x Multiplier
24,709 x 1.25 = 30,886.25

The festival ran for 8 weeks so:
30,886.25 x 8 = 247,090 tickets total – option D.

Explanation

Answer: C

Explanation: 

1. Find the coffee bean usage in cafés as a percentage of all the coffee beans used.

Beans in cafés: 242,000
Beans in total: 242000 + 50300 + 28341 + 36324 = 356965

242000/356965 = 0.6779… = 67.8%

Timing Tip: For this question, you could round the values to 50,000, 28,000 and 36,000 and you would still get an answer (67.9%) which clearly indicates option C

Timing Tip: Using rounded numbers, you could do some of these additions in your head or with the help of the scratch pad: 242 + 50 + 28 + 36 will tell you the number of thousands without having to type each number out in full.

Top Tip: Make sure to familiarise yourself with the scratchpad and calculator using the UCAT official website. You don’t want the first time you are using it to be in the real exam.

Explanation

Answer: C

Explanation: 

1. Find the sum of all the values in the table:

40000 + 25550 + 180000 + 30000 + 45243 + 25043 + 200,000 + 33000 + 48500 + 26030 + 220000 + 35050 + 50300 + 28341 + 242000 + 36324 + 55320 + 30245 + 266200 + 37200 = 1654346 Kg

2. Find the mean annual use:

The mean use = Total use in the period/Number of years

1654346/5 = 330869.2 so 330869 to the nearest kilogram.

Top Tip: Identify early on that this question is asking for the mean of all the values. Furthermore, the answer options are close together so it is clear that precision will be required. The best candidates will spot this is a potential timing trap and be strict about keeping to the 40s limit.

Explanation

Answer: A

Explanation:

A is incorrect. Between 2010-11 and 2011-12 the increase was 200,000/180,000 = 1.11 which is an 11% increase not 10%. The other values are 10% increases but not the first period.

B is true. It fell between 2010-11 and 2011-12 but rose every other year.

C is true. 50300/48500 = 1.0371… so a 3.7% increase

D is true. Only one value (restaurants in 2011-12) was lower than the year before. This decrease was more than compensated by the increases in values for the categories so this can be selected as true through inspection.

E is true. The percentage change is 10% in the first year, 6.2% in the second, 3.6% in the third and 2.4% in the final year.

Timing Tip: Options which only mention one of the categories should be targeted first – they only require the assessment of one subset of the data. This means D should be assessed last. From the others, it is clear that E would take the longest so should be a lower priority.

Explanation

Answer: D

Explanation: 

1. Find the mass of coffee used in each ratio:

90% was used at a ratio of 1:18:

0.9 x 200,000 = 180,000

10% was used at a ratio of 1:20
0.1 x 200,000 = 20,000

2. Find the volume of coffee produced:

Find the volume of coffee made from this mass:
1g = 1ml

180,000 was used in the ratio 1:18 so:
180,000 x 18 = 3,240,000Kg so 3,240,000L

20,000 was used in the ratio 1:20 so:
20,000 x 20 = 400,000Kg so 400,000L

3,240,000 + 400,000 = 3,640,000L – D.

Common Trap: Remember that not all of the coffee will be made in the 1:18 ideal ratio. If this were true, the answer would be C.

Explanation

Answer: D

Explanation: 

1. Find the total consumption from the old model:

40,000 + 25,550 + 180,000 + 30,000 = 275,550

2.  Find the new values:

New value / Multiplier = Old value:

For the private households:
30000/1.15 = 26,086.9…

For the cafés:
180000/0.89 = 202,247.191…

3. Find the new total and percentage difference:

40000 + 25550 + 202247… + 26086.9… = 293,883…
293883… /275,550 = 1.0665 = 6.7% increase

Common Trap:A 11% underestimate means that the value given is 89% of the true value (11% percentage decrease to reach it) so to calculate the true value, divide by 0.89 rather than multiply by 1.11.

Explanation

Answer: E

Explanation: 

1. Identify the data the question is asking for:

It has asked for the 2017/18 budget – this is not given directly in the question.
We are given the 2018/19 budget and how much more it is than the 2017/18 budget.

2. Take away the increases to find the previous budget:

17.9125 – 1.2 + 1.2 – 0.05 – 0.125 – 0.5 + 0.15 = 17.3875

Common trap: Remember to add on the values for when the budget has reduced from 2017/18 – two negatives make a positive.

Explanation

Answer: A

Explanation: 

1. Find the playing staff budget as a proportion of the known 2017/18 budget:

Proportion for playing staff = Playing staff budget / Total budget

The total budget for 2017/18 was 17.3875
The playing staff budget was: 8.5 – 1.2 = 7.3

7.3/17.3875 = 0.4198… = 42.0%

Explanation

Answer: D

Explanation: 

1. Find the wages in 2016/17

There was a 25% percentage decrease between 2016/17 and 2017/8:

Original Value x Multiplier = New Value
So
Wages in 2016/17 x 0.75 = Wages in 2017/18

Wages in 2017/18:
3 – -1.2 = 4.2 million

New Value / Multiplier = Original Value
4.2/0.75 = 5.6 million

Therefore, the bill in 2016-17 was £5.6 million – D

Common Trap: To calculate the original before a 25% decrease, divide by 0.75 rather than multiplying by 1.25.

Explanation

Answer: B

Explanation: 

1. Find the proportion of the budget spent on all types of staff:

Proportion on staffing = Staffing costs / Total budget:
Staffing cost:
8.5 + 3 + 1.5 + 0.275 = 13.275 million

Total budget = 17.9125 million

13.275/17.9125 = 0.7411 = 74.1%

Top Tip: Remember that proportions compare a single value in comparison to the total. Here, the value is £13.275 million in comparison to the total of £17.9125 million

Explanation

Answer: B

Explanation: 

1. Find the amount spent on marquee players and non-marquee players:

1.2.+ 12.7% of 8.5 million = £2.2795 million
8.5 – 2.1271 = £6.2205 million

2. Set up a ratio for the marquee players to non-marquee players:

2.2795: 6.2205

3. Simplify this ratio:

We know that one side is equal to one so divide through by 2.2795 as it must be the highest common factor:
6.2205/2.2795 = 2.73 = 3
Therefore, the simplified ratio is 1:3

Explanation

Answer: B

Explanation: 

1. Work out his rent in each of the years.

He paid 75% of the market average (which was £1000 PCM).
0.75 x 1000 = 750

The average price increased from 1000 to 1250.
To find the percentage increase use the formula:

New Value / Old Value = Multiplier
1250/1000 = 1.25 = 25%

His rent increased at double this rate so 50%.

For a percentage increase:
New value = Old value x multiplier

50% increase = multiplier of 1.5

New rent = 750 x 1.5
New rent = £1125

2. Find the difference from the market rent

The graph shows that the new average rent is £1250.

£1250 – £1125 = £125 so he pays £125 below the market rate.

Top Tip: Look over the Multiplier Method lesson on the Online Portal – getting to grips with it will save you time.

Explanation

Answer: B

Explanation: 

1. Work out the price he charges each tenant and so the income

The first tenants, he charges at the rate for 2015-16 – £2000 PCM

He does this for 24 months so: 2000 x 24 = £48,000

The second tenants, he charges at the new average rate for 2017-18 – £2250

Again, he does this for 24 months: 2250 x 24 = £54,000

54,000 + 48,000 = £102,000 – B.

Timing Tip: When adding numbers in the thousands, remember that you don’t have to write out every digit. 54 + 48 = 102 and is easier to do as mental maths or on the whiteboard.

Explanation

Answer: C

Explanation: 

1. Find the new rent

The initial rent was double the average market rate in 2017-18.

2500 x 2 = 5000.

He now charges 60% of that:

0.6 x 5000 = £3000.

2. Find the RTV

RTV = Monthly Rental Income / Property Value

RTV = 3000 / 600,000

RTV = 0.005 = 0.5% – option C.

Common Trap: Watch out when there are values which are different by a factor of 10 – it could be a sign that there is a common conversion error.

Explanation

Answer: C

Explanation: 

1. Find the monthly rent of each and answer by inspection:

In both years, she was paying half the rent.

The rent in 2015-16 for a two-bedroom was £1500 PCM – the same as for a
one-bedroom flat in 2018-19 so, with no further calculation, we can say she paid the same in both years.

Top Tip: Identifying questions which can be answered by inspection will save precious time by avoiding completely unnecessary calculations.

Explanation

Answer: C

Explanation: 

1. Find the percentage increase.

A percentage increase can be found using the formula:

He initially would have gone for a 1.45kg package (in the 1.25-1.5kg band) at standard delivery – $8.25.

He is now going for a 1.55kg package (in the 1.5-2.5kg band) with recorded delivery ($17.00)

This means that the percentage increase is 106%.

Top Tip: It is worth spending the time getting used to the multiplier method. Our research shows that it does save time

Explanation

Answer: B

Explanation: 

1. Find the percentage of parcels below 1.25kg.

The question information states that there were an equal number of small and large parcels. All of the small parcels (50% of the total) were less than 1.25kg.

From the large parcels – 14% were 1 – 1.25kg. This means that 14% of the 50% that were large were less than 1.25kg.

This can be calculated using decimals:

50% of 14% = 7%

The total which are less than 1.25kg is therefore:

50% + 7% = 57%.

Common Trap: It is important to see that the 14% of parcels between 1 – 1.25kg represent 14% of the 50% that are large parcels. Therefore, it only represents 7% of the total of parcels.

Explanation

Answer: D

Explanation: 

1. Find the new prices for each:

The new price can be found using the formula:

NewValue=OldValue×Multiplier

The percentage increase of 38% gives a multiplier 1.38:

For the standard postage:

9.50 x 1.38 = $13.11

For the recorded postage:

17.00 x 1.38 = $23.46

2. Find the difference for the old and the new prices

Old: 17.00 – 9.50 = $7.50

New: 23.46 – 13.11 = $10.35

Difference: 10.35 – 7.50 = $2.85 – option D

Explanation

Answer: B

Explanation: 

1. Find the number of parcels sold in the 1 – 1.25kg range.

X is the number of parcels sold which were in the 1 – 1.25kg range.

They sold 0.5X of these recorded.

They sold 0.5X of these standard.

752.50 = 0.5X x 14.25 + 0.5X x 7.25
752.50 = 10.75X
X = 752.50 / 10.75
X = 70 – this is 14% of the number of large parcels

2. Find the number in the range 500g – 1kg.

Let us call the total number of large parcels Y:

14% of Y = 70
0.14Y = 70
Y = 70/0.14
Y = 500

The question information states that there are an equal number of large and small parcels.

Therefore, there are 500 small parcels. 10% of small parcels are 500g – 1 kg.

1. x 500 = 50 – B

Common Trap: This question relied on noticing that the number of large parcels is equal to the number of small parcels. Always read the bullet points below tables in the scenario.

Explanation

Answer: B

Explanation: 

1. Find the first discount.

It is a 40% discount, so the new price is 0.6 times the full price.

2 Find the second discount.

It is a 10% discount so the price for paying upfront is 0.9 times the discounted price.

0.9 x 0.6 = 0.54 – 54%.

This means that the answer is B

Common trap: Note that the 10% discount is on the new price. This means the calculation is 0.6 x 0.9 rather than 0.6 – 0.1 = 0.5 or 50% discount.

Explanation

Answer: D

Explanation: The question has asked for the June value as a percentage of the May value.

1. Find the May and June values:

June: From the graph, this is roughly 1500.

May: From the graph, this is roughly 1000.

2. Find 1500 as a percentage of 1000.

New value / Old value = Multiplier.

1500/1000 = 1.5 = 150%.

Common Trap: Notice this has asked for the June value as a percentage of May rather than asking for the percentage increase.

Explanation

Answer: B

Explanation:

1. Find the average for each for the first quarter.
For many questions, this could be done by estimation through eyeballing the graphs to save time. However, this one has a scale which has too wide a margin of error for this.

For Coffee: (1000 + 1250 + 1750) / 3 = 4000/3
For Ice cream: (250 + 500 + 750) / 3 = 500

2. Find the average for each for the final quarter:
For Coffee: (1250 + 1500 + 1500) / 3 = 4250/3
For Ice cream: (750 + 250 + 100) / 3 = 1100/3

3. Find the difference in averages:
C: 4250/3 – 4000/3 = 83.3
I: 500 – 1100/3 = 133.3
133.3 – 83.3 = 50.

Timing Tip: Some QR questions would take longer than 40s to answer. Try to identify these and learn to be disciplined so that you are not spending 120s making sure that you get the answer right.

Explanation

Answer: D

Explanation:

1. Find the number of each type of ice cream sold:
1500 + 2000 = 3500 ice creams.
2/5ths are large so 0.4 x 3500 = 1400
3/5ths are small so 0.6 x 3500 = 2100

Top tip: This method is known as calculation by proportions – these come up in Lesson 6.

2. Find the revenue from each type of ice cream and thus the difference:

The large ice creams are £2.75:
1400 x 2.75 = £3850

The small ice creams are £2:
2100 x 2 = £4100

Find the difference between the two:
4100 – 3850 = £350

Explanation

Answer: B

Explanation:

1. Find the approximate coffee sales in the first quarter in 2018 and therefore the 2019 target.

1000+1250+1750 = 4000

The manager wants 5% growth.

4000 x 1.05 = 4200

2. Find the sales in January and February.

In 2018: 1000 + 1250 = 2250
In 2019: 2250 x 1.02 = 2295

3. Find the necessary number for March:

The cafe need to meet the target of 4200 so:

4200 – 2295 = 1905.

Top Tip: Graphs often have far more information than will be required to answer the question. Here, focus on the three bars which represent coffee sales in the first quarter.

Explanation

Answer: B

Explanation:

1. Find the number of each group who are left handed.

⅚ of the left-footed people are left-handed:
5/6 x 54 = 45

16.7% of the right-footed people are left-handed:
0.167 x 102 = 17

90% of those who expressed no preference are left-handed:
0.9 x 40 = 36

This means that the total is:
45+17+36 = 98

2. Find the total interviewed and thus the percentage who are right-handed:

40+54+102 = 196

If 98 are left handed:
196 – 98 = 98 right handed people

98/196 = 50%

Explanation

Answer: A

Explanation:

Top tip: The best method is to use a value of 100 individuals – this allows easy calculations from percentages.

1. Draw a Venn Diagram to represent the situation.
A + B + C + 30 = 100
A + B + C = 70

2. Form equations for the number who can do each of the activities and solve to find C then B.

Snowboarding: A + B = 43

Skiing: B + C = 34

Substitute the equation for snowboarding into the first equation to find the
value of C.
A + B + C + 30 = 100
A + B = 43
43 + C + 30 = 100
C = 27

Thus find the value of B
We know from the skiing group that:
B + C = 34
B + 27 = 34
B = 7% – A.

Explanation

Answer: B

Explanation:

1. Find the sum of the incomes and find the monthly average:

(120 + 145 + 130 + 15 +17.5 19 + 80 + 90 + 101) x 1000 = $717,500

Divide this by 3 to find the monthly average.
$717,500/3 = $239,166… = $239,250 to the nearest $250.

Common Trap: Remember that the question is looking for the average monthly income rather than the mean income from each revenue stream.

Explanation

Answer: B

Explanation:

1. Find the multiplier:

130/145 = 0.89655… = 10.345 % percentage decrease

To the nearest 0.1%, this is 10.3%

Top tip: The answer options are 0.1% away so it will be necessary to work precisely.

Top tip: The multiplier method will save time – get to grips with it in practice and you will save precious seconds.

Explanation

Answer: C

Explanation:

1. Find the total income from teaching:

120 + 145 + 130 = 395

2. Find the February revenue as a proportion of that:

145/395 = 0.367… = 37% to the nearest percentage point

Top tip: Proportion is the comparison of one characteristic with the whole group.

Explanation

Answer: D

Explanation:

1. Find the profit for each:

Teaching: 0.4 x 130,000 = $52,000
Writing: 0.52 x 101,000 = $52,520

2. Find the percentage difference between the two:
52,520/52000 = 1.01
Therefore, writing has a higher profit by 1%.

Common trap: The profit margin is different by 12% but this is not the same as the profit being different by 12%. The profit is also dependent on the level of revenue which is very different for the two revenue streams.