Unit 9
9 marks · NormalAmira is planning a trip and needs to exchange some money. In her country, £1 = $1.35. She exchanges £480 for dollars at this rate. After her trip, she has $216 left and wants to change this back to pounds at the same rate.
Calculate how many dollars Amira receives for her £480. Then, work out how many pounds she will receive when changing $216 back into pounds at the same rate.
Amira notices that the total number of pounds she gets back for her $216 is not in the same ratio as the pounds to dollars in the original exchange. Prove algebraically that, if no commission or rounding is involved and the rate is the same both ways, exchanging money and exchanging back always gives the original currency value (neglecting leftover cents or pence).
Work this out on paper, then compare with the answer and worked solution below. With a free account you can submit written answers for marking.
Show answer
(a) Amira receives $648 for £480 and £160 for $216
(b) If x units of original currency are exchanged at rate r then exchanged back at the inverse rate, the amount returns to x
Show mark scheme and worked solution
Marking points
M1 (a) · 1 mark · Correctly multiplies £480 by 1.35 to find total dollars received.
A1 (a) · 1 mark · Gets $648 as the amount received for £480.
M2 (a) · 1 mark · Divides $216 by 1.35 to find amount received in pounds.
A2 (a) · 1 mark · Gets £160 as the amount received for $216.
M3 (b) · 1 mark · Lets x be an amount exchanged at rate r, forming the algebraic expressions for exchange to foreign currency and back.
M4 (b) · 1 mark · Shows the use of inverse operation (dividing by r after multiplying by r, or shows rates are reciprocals).
A3 (b) · 1 mark · Reaches the algebraic conclusion that the value returns to x.
C1 (b) · 1 mark · Communicates reasoning clearly and logically in forming and explaining the algebraic proof.
B1 (b) · 1 mark · Makes and explains the required general statement, e.g. 'If the rate is exactly inverted, no commission, the process always retrieves the original sum.'
Worked solution
a) To change £480 into dollars at £1 = $1.35:
So Amira receives $648.
To change $216 back into pounds at the same rate:
So Amira receives £160.
b) Let x be an amount in pounds, and r be the exchange rate (£1 = $r).
When exchanging x pounds, she gets:
When exchanging back to pounds at the same rate:
So, without commission or rounding, exchanging and exchanging back at the same rate returns the original amount.