Money, Time and the Calculator

Money: two decimal places, and what the display is telling you

Money is written with two decimal places because there are 100 cents in a dollar and 100 pence in a pound. A calculator shows the shortest version of a number, so 4.8 on the display means 4.80 dollars and 12.5 means 12.50 dollars. Write every money answer with two decimal places, even when the second one is a zero.

The syllabus makes this explicit: interpret the display appropriately, because in money 4.8 means 4.80 dollars, while in time 3.25 means 3 hours 15 minutes. The same digits mean different things in the two contexts, and reading the wrong one loses a mark that has already been earned.

Round money to the nearest cent, which is two decimal places. $8\times 1.99=15.92$ exactly, but a calculation such as $11.50÷3=3.8333$ has to be rounded to 3.83 dollars. Only round at the end: if you round 3.8333 to 3.8 and then multiply by 6 you get 22.8, while the correct answer is 23.00 dollars, because $11.50÷3\times 6=23.00$ exactly.

Money questions also use prices given per unit. Fuel at 1.42 dollars per litre costs $35\times 1.42=49.70$ dollars for 35 litres, and apples at 2.35 dollars per kg cost $6\times 2.35=14.10$ dollars for 6 kg. The unit attached to the rate tells you whether to multiply or divide.

Money problems: totals, change and comparing prices

Most money questions take more than one step: work out each part, add the parts, then subtract from what was paid to find the change. Four notebooks at 2.85 dollars each come to $4\times 2.85=11.40$ dollars and three pens at 1.40 dollars each come to $3\times 1.40=4.20$ dollars, so the bill is $11.40+4.20=15.60$ dollars and the change from 20 dollars is $20-15.60=4.40$ dollars.

Work in the order the question gives you and write each step on its own line. One long line of calculator work that goes wrong cannot be awarded method marks, while three short lines can earn two method marks even if the final subtraction slips.

Comparing prices uses the same skill as best value in ratio and proportion: divide the cost by the quantity to get a price per unit. $5.20÷4=1.30$ dollars per kg against $7.50÷6=1.25$ dollars per kg, so the 6 kg bag is the better buy even though its total price is higher.

Bills often mix quantities. Four adults at 12.50 dollars and three children at 7.25 dollars comes to $50.00+21.75=71.75$ dollars. Keep every amount as money with two decimal places rather than converting to a whole number of cents, because one slip in the conversion costs the answer.

• Total cost: multiply the quantity by the price for each item, then add the results
• Change: subtract the total cost from the amount handed over
• Best buy: divide cost by quantity for each option and compare the unit prices
• Check the answer is sensible: change is always less than the amount paid

Converting from one currency to another

A currency question always gives you the rate, and you must use that rate rather than one you remember. At 1 pound = 1.25 dollars, 240 pounds is $240\times 1.25=300$ dollars and 450 dollars is $450÷1.25=360$ pounds.

Decide the direction from the size of the answer. If one pound is worth more than one dollar, a sum of pounds must become a bigger number of dollars, so you multiply; a sum of dollars must become a smaller number of pounds, so you divide. That single check catches most currency errors before they are written down.

Some rates are given the other way round, such as 1 dollar = 0.80 pounds. Then 150 dollars is $150\times 0.80=120$ pounds and 500 pounds is $500÷0.80=625$ dollars. Read which currency the 1 stands for, because it decides the whole calculation.

Converted money is still money, so round it to two decimal places. At 1 pound = 1.32 dollars, 85 pounds is $85\times 1.32=112.20$ dollars, and 200 dollars is $200÷1.32=151.5151$, which rounds to 151.52 dollars. If a question says the rate is approximate, that rounding is worth a comment when the question asks you to explain.

Time units and the relationships between them

The syllabus lists the units you have to handle: seconds, minutes, hours, days, weeks, months and years. Learn the fixed relationships: 60 seconds = 1 minute, 60 minutes = 1 hour, 24 hours = 1 day, 7 days = 1 week and 12 months = 1 year. Cambridge states that 1 year = 365 days, so use 365 rather than 365.25 unless the question says otherwise.

Multiplying those together gives the conversions questions actually use. One hour is $60\times 60=3600$ seconds, one day is $24\times 3600=86400$ seconds, one week is $7\times 24=168$ hours, and one year is $365\times 24=8760$ hours.

To add or subtract times, keep hours and minutes separate and convert whenever you pass 60 minutes. 1 hour 45 minutes plus 2 hours 30 minutes is 3 hours 75 minutes, and 75 minutes is 1 hour 15 minutes, so the total is 4 hours 15 minutes. Treating the minutes as a decimal, $1.45+2.30=3.75$, gives the wrong answer.

When a time is used in a rate or speed calculation, convert it to hours as a decimal fraction of an hour: 15 minutes is 0.25 hours, 20 minutes is $\frac{1}{3}$ of an hour, 30 minutes is 0.5 hours and 45 minutes is 0.75 hours. Times that are not multiples of 15 minutes do not give tidy decimals - 25 minutes is $\frac{5}{12}$ of an hour - which is a sign to keep the fraction rather than round early.

The 12-hour clock and the 24-hour clock

The 24-hour clock writes times as four digits with no a.m. or p.m.: 3.15 a.m. is 03 15 and 3.15 p.m. is 15 15. The hours run from 00 to 23, and a 24-hour time is read as hours then minutes.

To change an afternoon or evening time to the 24-hour clock, add 12 to the hour: 3.15 p.m. is $3+12=15$, so 15 15. Then 7.40 p.m. is 19 40 and 11.55 p.m. is 23 55. Morning times keep the same hour, written with two digits, so 6.50 a.m. is 06 50 and 9.05 a.m. is 09 05.

To go back, subtract 12 from any hour of 13 or more: 17 30 is 5.30 p.m., 13 05 is 1.05 p.m. and 21 40 is 9.40 p.m. Times from 00 00 up to 12 00 are a.m. times, except 12 00 itself, which is noon.

Midnight and noon are the two times that catch students out. Midnight is 00 00 on the 24-hour clock and 12 a.m. on the 12-hour clock; noon is 12 00 and 12 p.m. So 12.30 a.m. is 00 30, while 12.30 p.m. is 12 30.

Clocks, timetables and working out a length of time

A timetable question asks you to read two times and find the difference, or to add a journey time to a departure time. Keep the hours and the minutes separate and borrow or carry 60 minutes, never 100.

To find a journey length, subtract as you would with money but borrow 60 rather than 100. From 08 40 to 10 15: 10 15 is the same as 09 75, so 09 75 minus 08 40 is 01 35, a journey of 1 hour 35 minutes. The same method from 07 42 to 10 08 gives 2 hours 26 minutes.

To add, carry 60 minutes into an hour. A film starting at 20 20 that lasts 1 hour 55 minutes ends at 21 75, and 75 minutes is 1 hour 15 minutes, so the end time is 22 15. A train leaving at 08 45 with a journey of 2 hours 50 minutes arrives at 10 95, which is 11 35.

Timetables also give regular intervals. If a bus leaves every 25 minutes from 06 50, the departures are 06 50, 07 15, 07 40 and 08 05, and you can count how many fall in a given period by dividing the elapsed time by the interval and remembering the first departure.

• Subtract times by borrowing 60 minutes: $1015-0840=0975-0840=0135$
• Add times by carrying 60 minutes: 20 20 plus 1 hour 55 minutes is 21 75, which is 22 15
• Crossing midnight: 23 40 plus 7 hours 35 minutes is 06 40 the next day, then plus 35 minutes is 07 15
• A regular service: divide the elapsed time by the interval to count the departures

Time zones, local times and time differences

Different places keep different local times, and a question will tell you the difference. A city that is 3 hours ahead of another has a local time 3 hours later. Adding or subtracting that difference is the whole method.

If it is 11 30 in one city and the other city is 3 hours ahead, the local time there is $11+3=14$, so 14 30. If the other city is 4 hours behind and the time there is 22 50, then the local time here is $22-4=18$, so 18 50 the same day.

Flight questions combine a duration with a time difference, and the order matters. Add the flight time to the departure time in the departure's local time, then adjust for the time zone. A plane leaving at 09 20 local on a 5 hour 40 minute flight arrives at 15 00 in the departure's local time; if the destination is 3 hours ahead, the local time on landing is 18 00.

Watch for midnight. A plane leaving at 22 50 local on a 7 hour 20 minute flight arrives at 06 10 the next day in the departure's local time, and if the destination is 4 hours behind, that becomes 02 10 on that same calendar day. Write the day as well as the time whenever the answer crosses midnight, because a bare time is ambiguous.

Using a calculator efficiently, with money and time

The syllabus asks you to use a calculator efficiently, which means two things: do not round values within a calculation, and round only the final answer. Enter the whole expression in one go, using brackets where the order of operations would otherwise be wrong.

For a time question, enter 2 hours 30 minutes as 2.5 hours, or use the degrees, minutes and seconds key to enter 2 degrees 30 minutes 0 seconds. The syllabus gives both routes and they give the same answer. In the same way, 3 hours 15 minutes is 3.25 hours, and 1 hour 45 minutes is 1.75 hours.

Read the display in the context of the question. In a money question 4.8 is 4.80 dollars; in a time question 3.25 is 3 hours 15 minutes, not 3 hours 25 minutes. Turning a decimal answer back into hours and minutes means multiplying the part after the decimal point by 60: 4.6 hours is 4 hours plus $0.6\times 60=36$ minutes, so 4 hours 36 minutes.

Do not add or subtract clock times as decimals. $1.30+2.45$ typed into a calculator gives 3.75, but 1 hour 30 minutes plus 2 hours 45 minutes is 4 hours 15 minutes, because 75 minutes is an hour and a quarter. Convert to minutes, or keep hours and minutes separate, whenever the numbers are clock times.

Two habits save marks. First, keep the full display on screen, write down the unrounded value, and round once at the end. Second, check the answer is the size you expected: an average speed of 200 km/h for a bus, or a journey time of 3 minutes for a 200 km trip, means something went wrong in the entry.

• Enter 2 hours 30 minutes as 2.5, or as 2 degrees 30 minutes 0 seconds
• Enter 3 hours 15 minutes as 3.25, or as 3 degrees 15 minutes 0 seconds
• Round only at the end: $11.50÷3=3.8333$, which is 3.83 dollars
• Change a decimal answer back into hours and minutes by multiplying the part after the point by 60