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# Video Solutions – Test 1

Video Solutions – Test 12018-10-27T12:49:39+00:00

Question 1 In a box of 40 chocolates, 12 are made from white chocolate. What proportion of the chocolates are not made from white chocolate? Give your answer as a decimal.

Solution If 12 chocolates are made from white chocolate, then $40-12=28$ chocolates are not made from white chocolate. Thus, the proportion of chocolates that are not made from white chocolate is: $$\frac{28}{40}=\frac{7}{10}=0.7$$

Question 2 400 employees of the same company all claimed travel expenses. They each travelled 15 miles. Company expenses are paid at 30 pence per mile. In pounds, how much did this cost the company in total?

Solution 400 employees each travelled 15 miles, therefore they have travelled a combined mileage of $400 \times 15=6000$ miles. Each mile travelled costs the company 30 pence. Hence, the company pays $6000 \times 30=180000$ pence in total. However, the question asks for us to give our answer in pounds. Dividing our answer by 100 yields the answer £1800.

Question 3 In a football squad of 25 players, 5 of the players are above the age of 30. What percentage of the squad are older than 30?

Solution We need to find the proportion of players who are older than 30 years old. We know that 5 out of 25 players are older than thirty. Thus, we write:

$$\frac{5}{25}$$

Now we could divide 5 by 25 and then multiply the result by 100 to get the answer, but there is a quicker way of doing this. Notice that the denominator (bottom) of the fraction is 25. We know that $25 \times 4 = 100$. With fractions, as long as you multiply/divide the numerator and denominator of the fraction by the same number, then the fraction stays the same. Thus, we can multiply the top and bottom of our fraction by 4 to get.

$$\frac{5}{25}=\frac{5 \times 4}{25 \times 4}=\frac{20}{100}$$

Question 4 There were 30 children in the class. 250ml of apple juice was given to each child. How many 1 litre cartons of apple juice were needed?

Solution We first need to find the total volume of liquid required. To do this, we multiply the number 250 by the number of children in the class. This gives us:

$$250 \times 30 = 7500$$

So we need 7500ml, or 7.5litres of apple juice in total. Since the cartons contain exactly 1 litre of liquid, we need at least 8 apple juice cartons. Thus, the answer is 8.

Question 5 A family of four people wins £321 in a raffle. The family decides to share the money equally between each of the four family members. How much will each family member receive?

Solution The calculation we need to do is $321 \div 4$. We can compute this by either long/short division or by dividing by 2 twice to get the answer:

$$321 \div 4=80.25$$

Question 6 Two students both attend a sponsored walk to raise money for charity. The each walk 24 miles and are sponsored at 24 pence per mile. How much do they raise in total? Give your answer in pounds and pence.

Solution Each student raises 24 pence per mile that they walk. But each student walks exactly 24 miles, so they each raise $24 \times 24$ pence for charity. Since there are two students, we need to multiply this number by 2. Thus, the calculation we need to do is:

$$24 \times 24 \times 2 = 576 \times 2=1152$$

From the above, we can see that the pair raised 1152 pence between them. But the question asks for the answer in pounds and pence, so we need to divide our answer by 100 to get 11.52.

Question 7 What is 325 divided by 0.2?

Solution We need to find $325 \div 0.2$. Notice that $0.2=\frac{2}{10}$. So dividing by 0.2 is the same as dividing by $\frac{2}{10}$ (since $0.2=\frac{2}{10}$). But dividing by a fraction is exactly the same as multiplying by its reciprocal (the flipped fraction). Thus,

$325 \div 0.2$ $=325 \div \frac{2}{10}$ $=325 \times \frac{10}{2}$ $=\frac{3250}{2}$ $=1625$

Question 8 A couple wish to buy £400 worth of dollars for their weekend in New York. At the current rate, you can buy 1.25 dollars for £1. How many dollars will the company receive for £400?

Solution We consider the ratio $\pounds 1: \ 1.25$. Since we have £400, we need to multiply 1.25 by 400 to get our answer.

$$400 \times 1.25 =500$$

Hence, the answer is 500 dollars.

Question 9 An architect buys 21 pencils, each costing 12 pence. How much does this cost the architect in total? Give your answer in pounds.

Solution We simply compute $21 \times 12$. This yields

$21 \times 12 =252$ pence.

The questions asks for our answer to be given in pounds, so we divide our answer by 100 to get £2.52.

Question 10 A car dealership has 42 cars. If 28 of the cars are blue, what proportion of the cars are not blue? Give your answer as a decimal to two decimal places.

Solution If 28 of the cars are blue then $42-28=14$ of the cars are not blue. So we know that 14 out of a total of 42 cars are not blue. Thus, the proportion is

$$\frac{14}{42}=\frac{7}{21}=\frac{1}{3}$$

To get $\frac{1}{3}$ we have used the fact that multiplying/dividing the top and bottom of a fraction by the same number does not change the fraction. We first divided both sides by 2 to simplify the fraction, and then we spotted that we could also divide both sides by 7 to get the fraction in its lowest terms.

Now you might know off the top of your head that $\frac{1}{3}=0.\dot{3}=0.333333....$, but if you don’t then you would need to compute $1 \div 3$ using short division to get this result.

However, the question asks for the answer to be rounded to two decimal places. Thus, the answer is 0.33.

Question 11 Workers in an office are either analysts or supervisors. There are 48 analysts in the office. If supervisors manage a team of exactly six analysts and any two supervisors do not manage the same analysts, how many people are there in the office in total?

Solution The total number of workers in the office is going to be the total number of analysts (48) plus the total number of supervisors. We first need to calculate the total number of supervisors in the office. We know that supervisors manage a team of exactly six analysts. Thus, there are $48 \div 6 = 8$ supervisors in total.

Thus, the total number of workers is $48+8=56$.

Question 12 Sarah sends 15 text messages in five minutes. On average, how long does it take her to send one text message? Give your answer in seconds.

Solution If Sarah sends 15 text messages in 5 minutes then it takes her $5 \div 15 = \frac{5}{15}=\frac{1}{3}$ minutes to send one text. But $\frac{1}{3}^{\text{rd}}$ of a minute is $\frac{1}{3} \times 60= \frac{60}{3}=20$ seconds.

Hence, the answer is 20 seconds.

Alternative Solution Alternatively, we could first work out that five minutes is equal to 300 seconds and then calculate the following:

$$300 \div 15=\frac{300}{15}= \frac{100}{5}=20$$

Question 13 A class of ten A-Level maths students individually sat two test papers. The first paper was an algebra paper and the second was calculus. The percentage scores of both papers are shown on the scatter graph below. What proportion of the class scored higher in algebra than calculus? Give your answer as a decimal.

Solution There are ten students in total. We can confirm this by counting that there are a total of ten dots on the graph.

We need to first count the number of students who did better in algebra than in calculus. Notice that the $y=x$ line (grey line) has been added onto the graph. Points that lie exactly on the grey line indicate students who got exactly the same mark in both calculus and algebra. Points that are above the line represent students who performed better in calculus, and points below the line represent students who performed betting in algebra.

All we need to do, therefore, is to count the number of dots below the line; there are 5 in total.

Now all we need to do is to find the proportion in decimal form. There are 5 out of a total of 10, so we write:

$$\frac{5}{10}=0.5$$

Question 14

A company earns revenue from four electrical products that it invests in: A, B, C and D. Each of the four products has a set lifetime.

Which product generates the most revenue for the company throughout its lifetime?

Solution There are four different products. Each product has a set lifetime. For example, product A has a lifetime of 100 days. We are also given the revenue that each product generates per day. Again, as an example, we can see that product A generates £20 per day.

To answer this question, all we need to do is to calculate the total revenue generated by each of the four products throughout their lifespan. To do this, we multiply the lifetime (number of days) by the revenue per day for each product and then see which is the highest. This yields:

$100\times 20 =\pounds 2000$ $70 \times 25 = \pounds 1750$ $30 \times 30 = \pounds 900$ $115 \times 15 = \pounds 1725$ The largest of these quantities is £2000 (product A). Thus, our answer is A.

Question 15

The revenues over the past three years of two market competitors, Company A and Company B, are displayed below. Which of the following statements are true? A: The revenues of Company A doubled from 2014 to 2015 B: The revenues of Company B saw an increase of 50% from 2014 to 2015 С: There was a £20,000 difference in revenues between the two companies in 2015

Solution

Statement A We read from the bar chart that the revenues of Company A were £30000 in 2014 and £60000 in 2015. Since $60000=2 \times 30000$, we know that statement A is TRUE.

Statement B The revenues of Company B were £40000 in 2014 and £70000 in 2015.

$$40000\longrightarrow 70000$$

Let us increase the revenues of Company B in 2014 by 50% and see if this equals £70000. To increase a number by 50%, we multiply it by 1.5.

$$40000 \times 1.5=60000$$

So if the revenues increased by 50% from 2014 to 2015 then the revenue in 2015 would be £60000. But, we know from the graph that the 2015 revenues are actually £70000, so the statement is FALSE.

Statement C In 2015, Company A’s revenue was £60000 and Company B’s revenue was £70000.

$$70000-60000=10000$$

The difference between the 2015 revenues of the companies is £10000, not £20000, so the statement is FALSE.

Question 16

The HR manager of an organisation wishes to look into reducing work related stress in the workplace across the four different departments in the company. She produces the following table.

What percentage of the workers (across the four departments) suffer from work related stress? Give your answer to one decimal place.

Solution

$\text{Proportion of workers suffering from stress} = \frac{\text{Total number of workers suffering from stress}}{\text{total number of workers}}$.

We first need to find the total number of workers suffering from work related stress. We can do this by adding up the numbers in the third column:

$\text{Total number of workers suffering from stress} = 3+3+2+10=18$.

We now need to find the total number of workers. We do this by summing the numbers in the second column:

$\text{Total number of workers} = 12+25+14+51=102$.

So we have 18 out of a total of 102 workers that suffer from work related stress. Thus, the proportion is:

$$\frac{18}{102}=0.1764705...=17.64705... \%$$

The question asks us to round our answer to one decimal place. Thus, the answer is 17.6%.

Question 17

The table below illustrates the time taken (in minutes) for 50 students to complete a maths problem.

What is the mean amount of time spent on the question? Give your answer in seconds.

Solution

$\text{Mean amount of time taken}=\frac{\text{total number of minutes}}{\text{total number of students}}$.

We first need to find the total number of students. We are told in the question that there are 50 students, but we could also sum up the values in the second column instead.

$18+27+5=50\text{ students}$.

We now need to find the total number of minutes. We need to remember that this is a frequency table; we cannot just simply sum all the values in the first column. Instead, we need to think of it as follows: if 18 people took 1 minute, then they spent a combined total of $1 \times 18=18$ minutes; if 27 people took 2 minutes, then they spent a combined total of $2 \times 27=54$ minutes; and if 5 people took 3 minutes, then they spent a combined total of $3 \times 5=15$ minutes to complete the problem.

Hence, the total number of minutes is $18+54+15=87$. Now, we may evaluate the mean to be:

$$\frac{87}{50}=1.74\text{ minutes} = 1.74 \times 60\text{ seconds}=104.4\text{ seconds}$$

Since the question asks for our answer in seconds only, we round our answer to the nearest second to get 104.

Question 18

A driving test instructor monitored the number of minors his students were getting in their driving tests against the amount of practice lessons they had with him.

Which of the following statements are true? A: Three tenths of the students had less than 9 lessons. B: The highest number of minors a student got was 10. C: The median number of lessons had 1 minor

Solution

Statement A We can see from the graph that a total of three students has less than 9 lessons. We can see that one students had a total of 6 lessons, one had 7 lessons and one had 8 lessons. We now need to find out how many students there were in total. To do this, we count up all of the circles on the scatter graph: there are 11 in total. So, we have 3 out of a possible 11 students that had 9 lessons or less. Thus, the proportion is:

$$\frac{3}{11} \quad (=0.272727...)$$

Clearly $\frac{3}{11}\; (0.27...)$ does not equal $\frac{3}{10}\; (0.3)$, so the statement is FALSE.

Statement B The number of minors a student was awarded is dictated by the height of the point (y axis). The further to the right a point is, the more lessons that student had. The further to the top a point is, the more minors the student was awarded. Thus, we need to look for the highest dot on the graph. We can see that the highest dot on the graph represents the student who had only six lessons. That student got 10 minors. Hence, the statement is TRUE.

Statement C The median of a data set is given by:

$$\text{Median}=\frac{n+1}{2}^{\text{th}}\text{ number}$$

Where $n$ represents the number of data points. In this example, $n=11$. We get the number 11 by counting the number of dots on the graph.

Using the formula, our median is the

$$\frac{11+1}{2}=6^{\text{th}}\text{ number}$$

Now, we are asked to find the median number of lessons. Since the number of lessons is on the horizontal axis, we need to count the dots horizontally. Counting the dots horizontally from smallest to largest, we get that the median number of lessons is 10 lessons.

We now read from the graph that the point representing 10 lessons had one minor. Thus, the statement is TRUE.

Question 19

A medical research company wishes to test the effectiveness of two newly developed medical drugs: Drug A and Drug B. To do this, the company records the levels of effectiveness from two samples of 30 patients. The results are compiled in the table below.

Which of the following statements are true? A: 1/6 of the patients taking Drug A experienced ‘high’ effectiveness. B: The majority of patients in the sample for Drug B experienced ‘low’ effectiveness. C: ¼ of the total patients across both samples experienced ‘low’ effectiveness.

Solution

Statement A We read from the table that the total number of Drug A patients that experienced high effectiveness was 5. So we have that 5 out of a total of 30 people experienced high effectiveness. Thus, the proportion is:

$$\frac{5}{30}=\frac{1}{6}$$

Hence, statement A is TRUE.

Statement B We read from the table that only 7 Drug B patients out of 30 experienced low effectiveness. Thus, statement B is FALSE.

Statement C In total there are $30+30=60$ patients across both samples. In total, $5+7=12$ patients experienced low effectiveness. Hence, the proportion is:

$$\frac{12}{60}=0.2$$

But the statement claims that $\frac{1}{4}=0.25$ of the patients experienced low effectiveness, so the statement is FALSE.

Question 20

The percentage scores of a spelling test sat by year 7 pupils are displayed in the cumulative frequency diagram below:

Which of the following statements are true? A: The median mark is 60 B: The median mark is 30 C: Not one pupil got a score above 90

Solution

Statement A To find the median of a cumulative frequency diagram, we first need to find the total number of people. To do this, we simply read off the highest level that the line reaches. We can see from the diagram that the highest point of the line represents a cumulative frequency of 100. Thus, there are 100 people in the sample.

All we need to do now is to divide this number by two $\frac{100}{2}=50$ and then read off the number of marks that represents a cumulative frequency of 50.

We can see that a mark of 60 represents a cumulative frequency of 50. Thus, the median mark is 60.

Hence, statement A is TRUE.

Statement B We know from the above that the median is 60 so this statement is FALSE.

Statement C We can see from the graph that the line is flat between 90 and 100 marks. This means that not one person got a mark above 90, so statement C is TRUE.

Question 21

The revenues for three companies over a three-year period are shown in the bar chart below.

In which year(s) did Company B earn one third of the total revenues of all three companies combined?

Solution

To answer this question, we need to calculate the proportion of market share that Company B had compared to the total of all three companies.

2014 $\text{Total market share}=90$. $\text{Company B market share}=90-60=30$.

In 2014, Company B had a market share of 30 out of a total of 90. Thus, the proportion is:

$$\frac{30}{90}=\frac{1}{3}$$

2015 $\text{Total market share}=60$. $\text{Company B market share}=60-50=10$.

In 2015, Company B had a market share of 10 out of a total of 60. Thus, the proportion is:

$$\frac{10}{60}=\frac{1}{6}$$

2016 $\text{Total market share}=80$. $\text{Company B market share}=80-50=30$.

In 2015, Company B had a market share of 30 out of a total of 80. Thus, the proportion is:

$$\frac{30}{80}=\frac{3}{8}$$

The only year in which Company B had one third of the market share across all three companies was in 2014, so our answer is 2014.

Question 22

Some students in two Maths classes had to resit a test. Out of those who were resitting, some were required to resit one test whilst others had to resit both. The results are displayed in the pie charts below. If there are 25 people in class 1 and there are the same number of people resitting the Trigonometry test in both classes, how many students are there in class 2?

Solution

We are told that there are 25 students in class 1 and the pie chart tells us that 20% of the students in class 1 are resitting trigonometry. Thus, a total of $25 \times 0.2=5$ of the pupils in class 1 are resitting trigonometry.

But the questions tells us that the there are the same number of people resitting trigonometry in both classes, so we know that there are also 5 people resitting trigonometry in class 2.

In fact, we can tell from the pie chart on the right that 12.5% of people in class 2 are resitting trigonometry. This tells us that:

$$0.125 \times \text{ the number of students in class 2}=5$$

Hence, there are $\frac{5}{0.125}=40$ students in class 2. Thus, the answer to the question is 40.

Question 23

The marks awarded for a piece of English coursework for two different classes in a school are compared by using box and whisker plots, as shown below.

Which of the following statements are true? A: The median in class A is 30. B: The highest mark was in class B C: The range of marks is bigger in class A

Solution

Statement A We read off from the boxplot on the left that the median mark is 30, so the statement is TRUE.

Statement B The highest mark is class A was 40 and the highest mark in class B was 45. Hence, the statement is TRUE.

Statement C The range of marks in class A is $40-20=20$ and the range of marks in class B is $45-10=35$, so the statement is FALSE.

Thus, the answer to the question is A,B.

Question 24

The percentage market share of four companies, each operating in different markets, over a four year period is displayed in the table below.

Which companies saw a continual decrease in the percentage market share over the four years?

Solution To answer this question, all we need to do is to pick out the companies that had a decrease in market share each consecutive year.

Company A If we look at company A, we can see that its market share increased from 2013-2014, so this is not a contender.

Company B If we look at company B, we can see that its market share decreased from 2013-2014, 2014-2015, and 2015-2016.

Company C The market share of company C increased from 2013-2014, so this is not a contender.

Company D If we look at company D, we can see that its market share decreased from 2013-2014, 2014-2015, and 2015-2016.

Question 25

The bar chart below shows the percentages of students in years 7, 8 and 9 who have had at least one detention this year. 36 out of the 144 students in year 9 have had at least one detention.

How high should the bar chart for year 9 be? Give your answer as a percentage.

Solution All we need to do here is evaluate the proportion. We know that 36 out of the 144 students in year 9 have had at least one detention, so we evaluate the proportion as:

$$\frac{36}{144}=0.25=25 \%$$

Question 26

The summer mathematics examination grades of two different classes in a school are compared in the bar chart below.

Which of the following statements are true? A: 10% of the students across both classes achieved A*- A grades. B: The same number of students in class 1 and class 2 attained D – F grades. C: There are 50% more students in class 1 who got grades D – F than there are students in class 1 who got B – C grades.

Solution

Statement A We first need to calculate the total number of students across both classes. We do this by summing up the heights of each of the six bars:

$$\text{Total number of students}=5+3+15+23+20+14=80$$

Now if 10% of the total students achieved A*-A grades, this means that $80\times 0.1=8$ students were awarded A*-A grades across both classes. In fact, there are 5+3=8 students who achieved A*-A grades, so the statement is TRUE. Statement B The heights of the two bars above D-F are not equal, so the statement is FALSE.

Statement C There are 15 students in class 1 that got B-C grades and 20 students in class 1 who got D-F grades.

$$15 \longrightarrow 20$$

We need to find out whether or not a 50% increase of 15 gives us 20. To find this out, we may calculate:

$$15 \times 1.5 =22.5$$

So if the number of students who got D-F grades was 50% higher than the number of people who got B-C grades then there would be 22.5 students who got D-F grades. This isn’t true, so the statement is FALSE.

Thus, the answer to the question is A.

Question 27

Two pupils sat the same two tests and their marks were recorded. The tests were weighted as follows: test 1 had a 40% weighting and test 2 had a 60% weighting. The formula to calculate their final mark is given by:

Final mark = (test1mark x 0.4)+(test2mark x 0.6) Which student achieved the highest final mark?

Solution To answer this question, we simply substitute the values in the table into the formula given in the question:

$\text{Final mark}=(\text{test 1 mark}\times 0.4 )+(\text{test 2 mark} \times 0.6)$.

Pupil A $\text{Final mark}=(80\times 0.4 )+(75 \times 0.6)=77$.

Pupil B $\text{Final mark}=(70\times 0.4 )+(82 \times 0.6)=77.2$.

It is clear from the above that Pupil B achieved the highest final mark, so our answer is B.

Question 28

The number of hours 7 students spent on revision in the winter and summer terms were recorded. The results are displayed in the table below.

What fraction of the students spent more hours revising in the summer term than in the winter term?

Solution To begin, we need to go through each student individually and count the number of students who spent more hours on revision in the summer than they did in the winter. From the table, we can see that students A, C, F and G spent more hours on revision in the summer than they did in the winter.

There are 7 students in total. We have identified that 4 out of seven students spent more hours revising in the summer than in the winter. As a proportion, this is:

$$\frac{4}{7}$$