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4024/21

Mathematics (Syllabus D) 4024/21 — October/November 2016

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

11
questions
100
marks
150
minutes

Topics Number · Mensuration · Algebra and Graphs · Geometry · Trigonometry · Coordinate Geometry · +4 more

Q1NumberFree sample
(a)

In 2016, the price of a television is $1995.

(i)

Afzal pays the $1995 with a deposit of $399 and 12 equal monthly payments.
Calculate Afzal’s monthly payment.

$ ______

1M
DifficultyMedium-Easy
Worked solution

Approach

Subtract the deposit from the total price to find the amount still owed, then divide that amount by the 12 equal monthly payments.

Working

1995−399=15961995 - 399 = 1596

Amount to be paid each month:

159612=133\frac{1596}{12} = 133

Answer

$133\boxed{\$133}
Final answer

$133

Detailed explanation

Walkthrough

The television costs $1995 and Afzal pays $399 of that as a deposit. The deposit reduces the amount he still owes, so subtract $399 from $1995 to find the balance. This balance is then paid in 12 equal monthly payments, so dividing by 12 gives one monthly payment. Since 1995−399=15961995 - 399 = 1596 and 1596÷12=1331596 \div 12 = 133, each payment is $133.

Key Takeaways

This question uses the idea of a deposit plus instalments. It shows that a total can be split into an initial payment and a set of equal later payments. The skill is to identify which amount is divided into 12 parts, not the original price.

Common Mistakes

  • Dividing the whole $1995 by 12 and forgetting to subtract the deposit first.
  • Subtracting 12 from $1995 or confusing the number of payments with the amount of the deposit.

Things to Be Careful About

The final answer must be in dollars, so include the dollar sign. The 12 equal payments are for the remaining balance only, so the deposit must be removed before dividing.

Techniques used
subtract the deposit from the total pricedivide the remaining balance by the number of monthly payments
(ii)

What percentage of $1995 is $399?

______ %

1M
DifficultyEasy
Worked solution

Approach

Write $399 as a fraction of $1995 and multiply by 100. Since 399×5=1995399 \times 5 = 1995, the fraction simplifies to 15\frac{1}{5}.

Working

3991995×100=15×100=20\frac{399}{1995} \times 100 = \frac{1}{5} \times 100 = 20

Answer

20%\boxed{20\%}
Final answer

20%

Detailed explanation

Walkthrough

To find what percentage $399 is of $1995, write the part ($399) over the whole ($1995) and multiply by 100. Because 399×5=1995399 \times 5 = 1995, the fraction 3991995\frac{399}{1995} simplifies to 15\frac{1}{5}. Then 15×100=20\frac{1}{5} \times 100 = 20, so $399 is 20% of $1995.

Key Takeaways

A percentage is a fraction out of 100. The formula is part over whole multiplied by 100. Recognising simple factors can make the calculation easier.

Common Mistakes

  • Giving 5% instead of 20% after working out that $1995 is 5 times $399.
  • Writing the fraction upside down as 1995399\frac{1995}{399}.

Things to Be Careful About

The final answer needs a percentage sign. The order matters: $399 is the part and $1995 is the whole.

Techniques used
write the part over the wholemultiply the fraction by 100 to find a percentage
(iii)

The price of the television in 2016 is 5% more than the price in 2015.
Calculate the price in 2015.

$ ______

2M
DifficultyMedium-Easy
Worked solution

Approach

The 2016 price is 105% of the 2015 price, because it is 5% more. Let the 2015 price be xx. Reverse the 105% multiplier to find xx.

Working

1995=105100x1995 = \frac{105}{100}x

Divide by the multiplier:

x=1995105×100x = \frac{1995}{105} \times 100 x=19×100=1900x = 19 \times 100 = 1900

Answer

$1900\boxed{\$1900}
Final answer

$1900

Detailed explanation

Walkthrough

The 2016 price is 5% more than the 2015 price, so it is 105% of the 2015 price. Therefore the given $1995 is not the original price but the increased price. To undo a 105% increase, divide by 105 and multiply by 100, or divide by 1.05. 1995105×100=1900\frac{1995}{105} \times 100 = 1900, so the 2015 price was $1900.

Key Takeaways

This is a reverse percentage problem. An increase of 5% means new price = 105% of old price, so to find the old price divide by 1.05. It is not the same as taking 5% off the new price.

Common Mistakes

  • Calculating 5% of $1995 and subtracting it, which gives $1895.25. This is wrong because $1995 is already the increased price.
  • Multiplying by 0.95 instead of dividing by 1.05.

Things to Be Careful About

The mark scheme awards a method mark for 1995105\frac{1995}{105} (or equivalent), so show the reversal step clearly. The answer must be exact, $1900, and include the dollar sign. Do not round an intermediate percentage before reversing.

Techniques used
recognise that the given price is 105% of the originaldivide by the percentage multiplier to undo the increase
(b)

Afzal watched a programme that lasted 2 hours 53 minutes.
It ended at 01 15.

At what time did it start?

______

1M
DifficultyMedium-Easy
Worked solution

Approach

Write the end time on a 24-hour scale that starts the day before, so the subtraction does not cross midnight awkwardly. Then convert the result to a 12-hour time if needed.

Working

The programme ended at 01 15. Since it started before midnight, call this time 25 15:

01 15→25 1501\,15 \rightarrow 25\,15

Subtract the duration:

25 h 15 min−2 h 53 min=22 h 22 min25\,\text{h}\,15\,\text{min} - 2\,\text{h}\,53\,\text{min} = 22\,\text{h}\,22\,\text{min}

So the start time was 22 22, which is 10 22 pm.

Answer

22 22 (or 10 22 pm)22\,22 \text{ (or } 10\,22\text{ pm)}
Final answer

22 22 (or 10 22 pm)

Detailed explanation

Walkthrough

The programme ended at 01 15 and lasted 2 hours 53 minutes. Since it began before midnight, think of 01 15 as 25 15 on the previous day's 24-hour clock. Then subtracting 2 hours 53 minutes gives 25 h 15 min−2 h 53 min=22 h 22 min25\,\text{h}\,15\,\text{min} - 2\,\text{h}\,53\,\text{min} = 22\,\text{h}\,22\,\text{min}. On the 24-hour clock that is 22 22, which is 10 22 pm.

Key Takeaways

Time is measured in hours and minutes, with 60 minutes in an hour. Subtracting a duration that crosses midnight can be done by adding 24 hours to the end time first.

Common Mistakes

  • Subtracting 2 hours 53 minutes directly from 1 hour 15 minutes and getting a negative time or an impossible time.
  • Giving 10 22 am instead of 10 22 pm.
  • Writing 12 22 or 11 22 through incorrect borrowing.

Things to Be Careful About

The mark scheme accepts 22 22 or 10 22 pm. If you use a 12-hour clock, include pm. If using a 24-hour clock, write four figures with a space, e.g. 22 22.

Techniques used
write a time after midnight as a time on the previous day's 24-hour scalesubtract a duration in hours and minutesconvert between 24-hour and 12-hour clock
(c)

A company paid a quarter of a million dollars for an advertisement that lasted 38 seconds.

Calculate the cost, correct to the nearest hundred dollars, for each second of the advertisement.

$ ______

2M
DifficultyMedium-Easy
Worked solution

Approach

A quarter of a million dollars is $250000. Divide this total cost by the number of seconds, then round to the nearest hundred dollars.

Working

10000004=250000\frac{1000000}{4} = 250000

Cost per second:

25000038=10000004×38=6578.947…\frac{250000}{38} = \frac{1000000}{4 \times 38} = 6578.947\ldots

Round to the nearest hundred dollars:

6578.947…≈66006578.947\ldots \approx 6600

Answer

$6600\boxed{\$6600}
Final answer

$6600

Detailed explanation

Walkthrough

A quarter of a million dollars is 10000004=250000\frac{1000000}{4} = 250000 dollars. The advertisement cost $250000 and lasted 38 seconds, so the cost per second is 25000038\frac{250000}{38}. This is a rate: total cost divided by the number of seconds. It equals 6578.947..., and rounding to the nearest hundred dollars gives 6600, because the tens digit is 7.

Key Takeaways

A 'quarter of a million' means 250000. Money spent over a time can be converted into a cost per second by dividing. Rounding to the nearest hundred means looking at the tens digit.

Common Mistakes

  • Treating a quarter of a million as 25000 or 2500000.
  • Giving 6578.95 or 6579 instead of rounding to the nearest hundred dollars.
  • Stopping at the exact quotient when a rounded answer is asked for.

Things to Be Careful About

The mark scheme allows 10000004×38\frac{1000000}{4 \times 38} as the method, so show the division clearly. The answer must be to the nearest hundred dollars, $6600, not the exact quotient. Include the dollar sign.

Techniques used
convert quarter of a million into a numberdivide the total cost by the length in secondsround to the nearest hundred dollars
(d)

The programme showed an athlete running 100 metres, measured correct to the nearest metre.
The time the athlete took was 11.3 seconds, measured correct to the nearest 0.1 second.

Calculate the upper bound of the athlete’s average speed.

______ m/s\text{m/s}

2M
DifficultyMedium
Worked solution

Approach

Average speed is distance divided by time. To get the upper bound, use the largest possible distance and the smallest possible time.

Working

Distance upper bound:

100.5 m100.5\ \text{m}

Time lower bound:

11.25 s11.25\ \text{s}

Upper bound of average speed:

100.511.25=8.9333…=8.93 m/s (3 s.f.)\frac{100.5}{11.25} = 8.9333\ldots = 8.93\ \text{m/s} \ (3\text{ s.f.})

Answer

8.93 m/s\boxed{8.93\ \text{m/s}}
Final answer

8.93 m/s

Detailed explanation

Walkthrough

The distance is 100 m to the nearest metre, so the true distance is between 99.5 m and 100.5 m. The time is 11.3 s to the nearest 0.1 s, so the true time is between 11.25 s and 11.35 s. Speed is distance divided by time. To make speed as large as possible, use the largest distance (100.5 m) and the smallest time (11.25 s). Then 100.511.25=8.9333...\frac{100.5}{11.25} = 8.9333..., which is 8.93 m/s to 3 significant figures.

Key Takeaways

For a measurement rounded to the nearest unit, the upper bound is halfway above and the lower bound is halfway below. For a quotient, the upper bound is obtained by using the upper bound of the numerator and the lower bound of the denominator.

Common Mistakes

  • Using the lower bound of distance and upper bound of time for the upper bound speed.
  • Using 11.35 as the time, forgetting that a smaller time gives a larger speed.
  • Rounding 8.9333... to 8.9 instead of 8.93 when 3 significant figures are used.

Things to Be Careful About

The mark scheme gives a mark for using 100.5 or 11.25, so write both bounds before the division. The final answer should be 8.93 m/s and include the unit. Round only at the end of the calculation.

Techniques used
find the upper bound of the distance from the rounded valuefind the lower bound of the time from the rounded valuedivide the extreme distance by the extreme time to obtain an upper bound

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