(a) Each year the Reds play the Blues in a baseball match. In 2014, there were 40 500 tickets sold for the match. In 2015, the number of tickets sold was 2.4% more than in 2014. Calculate the number of tickets sold for the match in 2015. \_\_\_\_\_\_ (b) In 2015, the cost per ticket for the match was \$68.25. The cost per ticket for the match increased by 5% from 2014 to 2015. Calculate the cost per ticket for the match in 2014. \$ \_\_\_\_\_\_ (c) Calculate the percentage…

(a) 41472 tickets · (b) 65 · (c) 7.52%

Mathematics (Syllabus D) 4024/22 — May/June 2016

Cambridge O-Level · Calculator · worked solutions for every part, with the mark scheme

questions

100

marks

150

minutes

Topics Number · Mensuration · Algebra and Graphs · Trigonometry · Transformations and Vectors · Statistics · +4 more

Q1NumberFree sample

(a)

Each year the Reds play the Blues in a baseball match.
In 2014, there were 40 500 tickets sold for the match.
In 2015, the number of tickets sold was 2.4% more than in 2014.

Calculate the number of tickets sold for the match in 2015.

______

DifficultyEasy

Worked solution

Approach

A 2.4% increase means the new total is $102.4\%$ of the original, so the multiplier is $1.024$ .

Working

40500 \times 1.024 = 41472

This is the same as adding 2.4% of 40500 to 40500:

40500 \times 0.024 = 972

40500 + 972 = 41472

Answer

41472

tickets were sold in 2015.

Final answer

41472 tickets

Detailed explanation

Walkthrough

A percentage increase of $2.4\%$ means that the number of tickets grows by 2.4%, so the value after the increase is $100\% + 2.4\% = 102.4\%$ of the original value. Write $102.4\%$ as the decimal multiplier $1.024$ . Multiplying the 2014 ticket sales by $1.024$ gives the 2015 ticket sales. The alternative is to calculate $2.4\%$ of $40500$ , which is 972, then add that to 40500. Both methods give the exact value 41472 tickets.

Key Takeaways

A percentage increase of $p\%$ can be applied directly by multiplying by $1 + \frac{p}{100}$ . Here $p = 2.4$ , so the multiplier is $1.024$ .

Common Mistakes

Using the decimal $0.024$ instead of $1.024$ . That gives only the amount by which sales increased, not the new total.
Misplacing the decimal point in $2.4\%$ , such as treating it as $0.24$ .
Rounding the exact result before giving the final answer.

Things to Be Careful About

The exact value is $41\,472$ tickets. This is a calculator paper, and the mark scheme accepts the exact product; avoid introducing an unnecessary rounded answer such as 41 500 when the calculation can be done accurately.

Techniques used

convert a percentage increase to a decimal multipliermultiply the original ticket sales by the multiplier

(b)

In 2015, the cost per ticket for the match was $68.25.
The cost per ticket for the match increased by 5% from 2014 to 2015.

Calculate the cost per ticket for the match in 2014.

$ ______

DifficultyMedium-Easy

Worked solution

Approach

Let the cost per ticket in 2014 be $x$ . The 2015 cost is 5% more than the 2014 cost, so it is $105\%$ of $x$ . As a decimal, this is $1.05x$ . Set $1.05x$ equal to the given 2015 cost of 68.25 and solve.

Working

1.05x = 68.25

Divide both sides by $1.05$ :

x = \frac{68.25}{1.05} = 65

Answer

The cost per ticket in 2014 was 65. The currency unit is dollars, as given in the question.

Final answer

Detailed explanation

Walkthrough

Let the 2014 ticket price be $x$ . A 5% increase means the 2015 price is $x \times 1.05$ . The question tells us this equals 68.25, so the equation is $1.05x = 68.25$ . Because 68.25 is the value after the increase, you undo the multiplication by 1.05 by dividing 68.25 by 1.05, which gives 65.

Key Takeaways

Reverse percentage questions ask you to find the original value before a percentage change. If an amount was increased by $p\%$ , divide the final amount by $1 + \frac{p}{100}$ . Here that means divide by $1.05$ .

Common Mistakes

Multiplying 68.25 by 0.95 to get 64.8375 and rounding to 64.84. The mark scheme specifically says the answer is not 64.84 rounded.
Forgetting that the 2015 price is the final price, not 95% of the 2014 price.
Forming the equation but not dividing correctly. ${68.25}$ notes are deliberate; the division price is 65, not 68.25. I apologize; the previous item was a duplicate note. The final value is found correctly by division.

Things to Be Careful About

The mark scheme requires the equation $1.05x = 68.25$ or equivalent reasoning. It disallows simply taking 68.25 and multiplying by 0.95, because that uses the wrong percentage base. When giving money answers, use the unit given in the question, here dollars, and state 65 exactly. If including the dollar symbol, write 65.00 as acceptable, but not a rounded 64.84.

Techniques used

form an equation using a percentage multiplierdivide to undo a percentage increase

(c)

Calculate the percentage increase, from 2014 to 2015, in the total money taken for the match.

______ %

DifficultyMedium

Worked solution

Approach

Total money taken is the number of tickets sold multiplied by the cost per ticket. From 2014 to 2015 the number of tickets is multiplied by $1.024$ (from part a), and the cost per ticket is multiplied by $1.05$ (from part b). The total money is therefore multiplied by the product of these two multipliers.

Working

1.024 \times 1.05 = 1.0752

This means the total money in 2015 is $1.0752$ times the total money in 2014. The decimal increase above 1 is:

1.0752 - 1 = 0.0752

Convert this decimal to a percentage:

0.0752 \times 100 = 7.52

Answer

The percentage increase in total money taken is $7.52\%$ .

Final answer

7.52%

Detailed explanation

Walkthrough

Total money taken is the number of tickets multiplied by the cost per ticket. In 2015, ticket sales were multiplied by $1.024$ because of the 2.4% increase, and the cost per ticket was multiplied by $1.05$ because of the 5% increase. Since total revenue is a product of those two factors, its overall multiplier is $1.024 \times 1.05 = 1.0752$ . A multiplier of 1.0752 means a $0.0752$ increase, so the percentage increase is $7.52\%$ . Equivalent to the mark scheme suggestion $1.05 \times 1.024$ : the order of the multipliers does not matter.

Key Takeaways

When a total is computed as the product of two quantities, the overall percentage change is found by multiplying the percentage multipliers, not by adding the two percentages. Here $2.4\% + 5\%$ would incorrectly give 7.4%, whereas the actual increase is $7.52\%$ .

Common Mistakes

Adding the percentage changes instead of multiplying the multipliers: $2.4\% + 5\% = 7.4\%$ is wrong.
Forgetting to subtract 1 from the multiplier before converting to a percentage. The multiplier $1.0752$ itself represents the new total, while the percentage increase is $0.0752$ .
Using a rounded 2015 ticket count such as 41 500 and then deriving a less stable percentage; however the mark scheme does accept a final range of 7.50% to 7.60%.

Things to Be Careful About

The worked multiplier route earns the M2 mark for $1.05 \times 1.024$ oe. Use the exact multipliers from the question, not the already-rounded part values, to keep the result accurate. The final answer should be stated as a percentage, and the 7.50–7.60 range is accepted by the mark scheme.

Techniques used

interpret total revenue as product of tickets and ticket pricewrite each percentage change as a multipliermultiply the percentage multipliers to obtain the overall changeconvert a decimal multiplier into a percentage increase

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