C1.16, C1.12; E1.16, E1.12 - Money and exchange rates

C1.16, C1.12; E1.16, E1.12 - Money and exchange rates

Money calculations need both accurate arithmetic and the correct currency. You will calculate costs and change, convert currencies in both directions and decide how to round a payment. C1.16 and E1.16 provide the money skills; C1.12 and E1.12 connect exchange rates to proportional reasoning.

Calculations with money

An exchange office offers 11 euro for \1.25.Ifyouchange. If you change 80$ euros into dollars, will you multiply or divide? A rate gives the value of one unit, just as a shop gives the price of one item. We will return to this after practising costs and change.

In a currency with 100100 cents to one dollar, \2.40meansmeans2dollarsanddollars and40cents.Youcancalculateindollarsusingdecimalsorinwholecents,butkeeponeunitthroughoutacalculation.Writeafinalmoneyamounttotwodecimalplaceswhenitincludesdollarsandcents:cents. You can calculate in dollars using decimals or in whole cents, but keep one unit throughout a calculation. Write a final money amount to two decimal places when it includes dollars and cents:$4.50,ratherthan, rather than $4.5$.

For identical items, multiply the unit price by the number of items. For a mixed basket, find each subtotal and add. Change is the amount paid minus the total cost; an answer below zero means the payment was insufficient.

Worked example 1: total cost and change

Question. Three notebooks cost \2.40each.Calculatethechangefromeach. Calculate the change from$10$.

Plan. Find the cost of all three notebooks, then subtract it from the payment.

3×2.40=7.20total cost in dollars10.00−7.20=2.80payment minus cost\begin{aligned} 3\times2.40&=7.20 && \text{total cost in dollars}\\ 10.00-7.20&=2.80 && \text{payment minus cost} \end{aligned}

Answer. The change is \2.80$.

Check. 7.20+2.80=10.007.20+2.80=10.00, so the cost and change account for the full payment.

Change only the number of notebooks. The unit price and payment stay the same.

Worked example 2: one more item

Question. Four notebooks cost \2.40each.Calculatethechangefromeach. Calculate the change from$10$.

Plan. Multiply by four, then subtract the cost from 1010.

4×2.40=9.60total cost in dollars10.00−9.60=0.40payment minus cost\begin{aligned} 4\times2.40&=9.60 && \text{total cost in dollars}\\ 10.00-9.60&=0.40 && \text{payment minus cost} \end{aligned}

Answer. The change is \0.40$.

Check. The extra notebook costs \2.40,sochangefallsfrom, so change falls from $2.80toto$0.40.Also,. Also, 9.60+0.40=10.00$.

The decimal point must stay aligned when you add or subtract money amounts.

Watch out

\0.40meansmeans40cents,notcents, not4cents.Writingcents. Writing$0.04$ would change the value. Final zeros after the hundredths place are unnecessary, but the hundredths place itself matters for money.

Try the same two-step calculation with a new price.

Converting currencies

An exchange rate relates equal values in two currencies. If £1=\1.20,eachpoundproduces, each pound produces 1.20dollars.Multiplypoundsbydollars. Multiply pounds by1.20togetdollars.Toreversetheconversion,dividedollarsbyto get dollars. To reverse the conversion, divide dollars by1.20$.

Think of a double number line: £1£1 matches \1.20,so, so £10matchesmatches$12andand£100matchesmatches$120$. Scaling both sides by the same number preserves the exchange rate. The rates in this lesson are the rates supplied by the problems, with no commission unless stated.

Worked example 3: converting in the quoted direction

Question. The exchange rate is £1=\1.20.Calculatehowmanydollarsarereceivedfor. Calculate how many dollars are received for £150$.

Plan. Multiply the number of pounds by the dollars received for one pound.

150×1.20=150+30one dollar and twenty cents per pound=180amount in dollars\begin{aligned} 150\times1.20&=150+30 && \text{one dollar and twenty cents per pound}\\ &=180 && \text{amount in dollars} \end{aligned}

Answer. \180$.

Check. £100£100 gives \120andand£50givesgives$60;togetherthesegive; together these give $180$.

Now reverse only the direction. The exchange rate itself has not changed.

Worked example 4: reversing the conversion

Question. The exchange rate is still £1=\1.20.Calculatehowmanypoundsarereceivedfor. Calculate how many pounds are received for $180$.

Plan. Find how many groups of \1.20fitintofit into$180$.

180÷1.20=1800÷12scale dividend and divisor equally=150one pound per group\begin{aligned} 180\div1.20&=1800\div12 && \text{scale dividend and divisor equally}\\ &=150 && \text{one pound per group} \end{aligned}

Answer. £150£150.

Check. 150×1.20=180150\times1.20=180, so converting back returns the supplied dollars.

The operations reverse because multiplication and division by the same nonzero rate undo each other.

Watch out

“Convert currency” does not always mean multiply. 180×1.20=216180\times1.20=216 is wrong for dollars to pounds with this quote. First identify which currency one quoted unit buys.

Keep the quoted direction clear in this check.

Checking exchange-rate direction and precision

The currency names alone do not tell you whether the numerical amount should rise or fall. Test your method on the quoted one unit. If \1=0.8euros,euros,$10mustgivemust give8$ euros: multiplication can make the numerical amount smaller.

Keep all the digits of the given exchange rate. Unless the question specifies something else, round the final payment to the nearest cent. Intermediate rounding changes the agreed rate and can change the payment. If a commission is stated, apply it exactly as described; do not assume a fee when none is given.

Worked example 5: keeping a rate intact

Question. An exchange office uses £1=1.125£1=1.125 euros. Calculate the euros received for £72£72, giving the answer to the nearest cent.

Plan. Use the full rate, then write the final amount to two decimal places.

72×1.125=72×98use an exact equivalent fraction=9×9divide by eight first=81amount in euros\begin{aligned} 72\times1.125&=72\times\frac98 && \text{use an exact equivalent fraction}\\ &=9\times9 && \text{divide by eight first}\\ &=81 && \text{amount in euros} \end{aligned}

Answer. 81.0081.00 euros.

Check. 1.1251.125 is 1+181+\frac18. One eighth of 7272 is 99, and 72+9=8172+9=81.

Change only the number of pounds by one. This time the exact result includes a fraction of a cent.

Worked example 6: rounding the payment

Question. At the same rate, calculate the euros received for £73£73, to the nearest cent.

Plan. Multiply using the complete rate, then round the answer to two decimal places.

73×1.125=82.125exact conversion before rounding82.125⟶82.13round the third decimal place\begin{aligned} 73\times1.125&=82.125 && \text{exact conversion before rounding}\\ 82.125&\longrightarrow82.13 && \text{round the third decimal place} \end{aligned}

Answer. 82.1382.13 euros.

Check. £72£72 gave 8181 euros, and the extra pound adds 1.1251.125 euros: 81+1.125=82.12581+1.125=82.125. The final rounding changes this by only 0.0050.005 euros.

When comparing offers, convert the same starting amount into the same destination currency at each office, including any stated fees. Comparing unlike quotes without reversing one of them can be misleading.

Watch out

Rounding 1.1251.125 to 1.131.13 before converting £72£72 gives 81.3681.36 euros. That uses a different rate. Keep the supplied rate intact and round only the payment.

Now make a fresh conversion at that rate.

Quick Recap

  • Use one money unit consistently; cost is quantity multiplied by unit price, and change is payment minus cost.
  • Read an exchange-rate quote as equal values and scale both sides together.
  • Reverse a conversion by undoing the multiplication with division.
  • Keep the stated rate intact and round the final payment as requested.

In the opening problem, each euro buys \1.25,so, so 80eurosbuyseuros buys80\times1.25=$100$. Dividing would reverse the intended conversion.