(a) The rate of exchange between dollars (\$) and pounds (£) is \$1.56 = £1. The rate of exchange between euros (€) and pounds is €1.10 = £1. (i) Amy changes £300 into dollars. Calculate how many dollars Amy receives. \$ \_\_\_\_\_\_ (ii) Ben changes €770 into pounds. Calculate how many pounds Ben receives. £ \_\_\_\_\_\_ (iii) Chris changes \$780 into euros. Calculate how many euros Chris receives. € \_\_\_\_\_\_ (b) Debbie changed some dollars into Japanese yen. The rate…

(a)(i) \$468 · (a)(ii) £700 · (a)(iii) €550 · (b) x = 19926

Mathematics (Syllabus D) 4024/21 — October/November 2013

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

questions

100

marks

150

minutes

Topics Number · Algebra and Graphs · Geometry · Transformations and Vectors · Trigonometry · Coordinate Geometry · +3 more

Q1NumberAlgebra and GraphsFree sample

(a)

The rate of exchange between dollars ($) and pounds (£) is $1.56 = £1.
The rate of exchange between euros (€) and pounds is €1.10 = £1.

(i)

Amy changes £300 into dollars.

Calculate how many dollars Amy receives.

$ ______

DifficultyEasy

Worked solution

Approach

The rate $1.56 = £1 means each pound is worth 1.56 dollars, so multiply the number of pounds by 1.56.

Working

300 \times 1.56 = 468

So Amy receives $468.

Answer

$468

Final answer

$468

Detailed explanation

Walkthrough

The exchange rate $1.56 = £1 tells you how many dollars each pound becomes. Amy has £300, so multiply 300 by 1.56.

300 \times 1.56 = 468

The answer is 468 dollars.

Key Takeaway

When an exchange rate is given as the amount of a currency per pound, multiply the number of pounds by that rate to convert into the other regulate.

Common Mistakes

Dividing 300 by 1.56. That would be the count of pounds you could buy with 300 dollars, which is the reverse conversion.
Rounding 1.56 to 1.6 before multiplying, losing the exact value.

Things to Be Careful About

The answer must be given in dollars, $468. Since this is the calculator paper, multiplying the decimals exactly is enough; no rounding instruction is given.

Techniques used

read the exchange rate in the correct directionmultiply the amount of pounds by the dollar rate

(ii)

Ben changes €770 into pounds.

Calculate how many pounds Ben receives.

£ ______

DifficultyEasy

Worked solution

Approach

Since €1.10 = £1, €1 is worth $\frac{1}{1.10}$ pounds. To convert €770 into pounds, divide by 1.10.

Working

770 \div 1.10 = 700

So Ben receives £700.

Answer

£700

Final answer

£700

Detailed explanation

Walkthrough

The rate €1.10 = £1 tells how many euros make one pound. To change euros into pounds, each euro must be converted at $\frac{1}{1.10}$ pounds. So:

770 \div 1.10 = 700

Ben receives £700.

Key Takeaways

When the rate tells you how many euros equal one pound, dividing the number of euros by that rate converts it into pounds.

Common Mistakes

Multiplying €770 by 1.10. That would convert pounds into euros, not euros into euros.
Writing 699 or 691 because of early rounding; the division is exactly 700.

Things to Be Careful About

The answer is £700, not 699. Check: £700 at €1.10 = £1 gives $700 \times 1.10 = 770$ euros, which matches the question.

Techniques used

convert a foreign currency back to poundsdivide the given amount by the exchange rate

(iii)

Chris changes $780 into euros.

Calculate how many euros Chris receives.

€ ______

DifficultyMedium-Easy

Worked solution

Approach

The two rates are both against £1, so $1.56 and €1.10 are equal in pound value. Therefore the direct dollar-to-euro factor is $\frac{1.10}{1.56}$ . Multiply the $780 by this factor.

Working

780 \times \frac{1.10}{1.56} = 780 \times \frac{110}{156} = 550

Chris receives €550.

Answer

€550

Final answer

€550

Detailed explanation

Walkthrough

Both exchange rates compare with £1. Thus the same amount pounds that is worth $1.56 is also worth €1.10. So, to change dollars directly into euros, multiply the dollar amount by $\frac{1.10}{1.56}$ . Using that factor:

780 \times \frac{1.10}{1.56} = 550

Chris receives €550.

Key Takeaways

When two currencies are both quoted against a common currency, a direct exchange factor can be written as a fraction of the two rates. This avoids converting through the common currency in two separate steps.

Common Mistakes

Using $\frac{1.56}{1.10}$ which would convert euros into dollars, not dollars into euros.
Writing a factor that uses the wrong order; the euro rate must go in the numerator because you are converting dollars to euros.

Things to Be Careful About

In the working, the direct factor $\frac{1.10}{1.56}$ is the key step. Show it before processing the final multiplication; it is the factor required for full marks. The exact result is 550 euros.

Techniques used

build a direct conversion factor between dollars and euroscombine two exchange rates through a common base

(b)

Debbie changed some dollars into Japanese yen.
The rate of exchange was 81 dollars = 1 yen.

Emma changed the same number of dollars into yen.
The rate of exchange for Emma was 82 dollars = 1 yen.

Emma received 3 fewer yen than Debbie.

Given that the number of dollars changed each time is $x$ , find $x$ .

______

DifficultyMedium

Worked solution

Approach

Let $x$ be the number of dollars changed. At 81 dollars = 1 yen, Debbie receives $\frac{x}{81}$ yen. At 82 dollars = 1 yen, Emma receives $\frac{x}{82}$ yen. Since Emma receives 3 fewer yen than Debbie, subtract Emma's amount from Debbie's amount and set the result equal to 3.

Working

\frac{x}{81} - \frac{x}{82} = 3

Combine the fractions using the common denominator $81 \times 82 = 6642$ :

\begin{aligned} \frac{82x - 81x}{6642} &= 3 \\[4pt] \frac{x}{6642} &= 3 \\[4pt] x &= 3 \times 6642 = 19926 \end{aligned}

Answer

$x = 19926$

Final answer

x = 19926

Detailed explanation

Walkthrough

At 81 dollars for 1 yen, the money received is $\frac{x}{81}$ yen. At 82 dollars for 1 yen, the money received is $\frac{x}{82}$ yen.

Because 82 dollars per yen is a larger cost per yen, Emma receives fewer yen than Debbie. The question says Emma received 3 fewer yen, so the difference is:

\frac{x}{81} - \frac{x}{82} = 3

This is a fractional linear equation. Put the two fractions over the common denominator $81 \times 82 = 6642$ :

\frac{82x - 81x}{6642} = 3

So $\frac{x}{6642} = 3$ , and $x = 3 \times 6642 = 19926$ .

Key Takeaways

Exchange rates involving dollars per yen can be written as fractions: if $81$ dollars give $1$ yen, then $x$ dollars give $\frac{x}{81}$ yen.
A worded gap of "3 fewer" translates into a difference equation.
Fractional equations are solved by putting fractions over a common denominator.

Common Mistakes

Reversing the equation and writing $\frac{x}{82} - \frac{x}{81} = 3$ . Since $\frac{x}{81}$ is the larger quantity, the positive difference is $\frac{x}{81} - \frac{x}{82}$ .
Dividing by $5$ instead of adding or subtracting after getting the difference.
Using $x \times 81$ instead of $x/81$ for yen received.

Things to Be Careful About

The correct amount is x = 19926, not a rounded decimal.
The mark scheme gives the M2 mark for the correct difference equation; the two fractions $\frac{x}{81}$ or $\frac{x}{82}$ are alone worth a B1 mark.
Show both fractions and the combination of the fractions, so the method and follow-through marks are clearly visible.

Techniques used

convert each dollar amount into yen using a fractional rateform an equation from the difference between the two amountssolve a fractional equation

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