Mathematics notes, unit 4
Time
Textbook Mathematics Year 5 (DLP), Part 1 and Part 2, pages Part 1, 123–134; Part 2, 135–176
What this chapter covers
- You can find a duration in days and hours, and in days between two dates, using the number of days in each month and leap years (4.1.1)
- You can convert hours, days, years, decades and centuries to smaller units, when the amount is a whole number, a fraction or a decimal (4.2.1, 4.2.2)
- You can add time given in fractions, decimals and two units, and regroup the answer (4.3.1, 4.3.2)
- You can subtract time given in fractions, decimals and two units, and borrow when you need to (4.3.1, 4.3.2)
- You can solve problems about time, such as shelf life and ages, by understanding, planning, solving and checking (4.4.1)
4.1Duration of time
The duration is how long something takes: from the time it starts to the time it ends. In Year 5 the duration can be in days and hours, in days between two dates, or even across more than one year.
Duration = end time − start time. Always say the unit: "2 days 2 hours", "37 days". When you count the days from one date to another, the first day is counted too (add 1).
Days and hours
A study tour starts at 10:50 a.m. on 25 July 2021 and ends at 12:50 p.m. on 27 July 2021. Count whole days first, then the extra hours.
Situation based on Mathematics Year 5 (DLP), p.123.
Worked example: 24-hour clock
Find the duration from 1650 hours on Monday to 0550 hours on Friday.Step 1. Monday 1650 hours to Friday 1650 hours is 4 days.
Step 2. But the end time is 0550 hours, which is 11 hours earlier than 1650 hours (1650 − 0550 = 11 hours).
Step 3. 4 days − 11 hours = 3 days 24 hours − 11 hours = 3 days 13 hours.
Months and days
To find the duration in days between two dates, you must know how many days each month has.
| Months | Days |
|---|---|
| January, March, May, July, August, October, December | 31 |
| April, June, September, November | 30 |
| February in a regular year | 28 |
| February in a leap year | 29 |
A leap year has 366 days. A year such as 2020 is a leap year because 2020 ÷ 4 = 505 with no remainder. A leap year comes once every 4 years.
Worked example: 1 February 2020 to 8 March 2020
Step 1. February 2020 is a leap year, so 1 February to 29 February = 29 days.Step 2. 1 March to 8 March = 8 days.
Step 3. Total = 29 + 8 = 37 days.
The same dates in 2019 (not a leap year) give 28 + 8 = 36 days.
Worked example: 29 May 2020 to 5 July 2020
Step 1. 29 May to 31 May = 31 − 29 + 1 = 3 days.Step 2. All of June = 30 days.
Step 3. 1 July to 5 July = 5 − 1 + 1 = 5 days.
Step 4. Total = 3 + 30 + 5 = 38 days.
Add the extra 1 day. The first day is also counted. 29 May to 31 May is 3 days (29, 30, 31), not 31 − 29 = 2 days.
Years, months and days
Worked example: 1 December 2019 to 19 January 2021
Step 1. 1 December 2019 to 31 December 2019 = 31 days.Step 2. The whole of 2020 (a leap year) = 366 days.
Step 3. 1 January 2021 to 19 January 2021 = 19 days.
Step 4. Total = 31 + 366 + 19 = 416 days.
Find the duration, in days, from 5 February 2020 to 3 March 2020.
Show answerHide answer
5 to 29 February = 29 − 5 + 1 = 25 days. 1 to 3 March = 3 days. Total = 28 days.
4.2Converting units of time
To change a bigger unit to a smaller unit, multiply. Use the same multipliers for whole numbers, fractions and decimals.
| Convert | Rule |
|---|---|
| hours to minutes | hours × 60 |
| days to hours | days × 24 |
| years to months | years × 12 |
| decades to years | decades × 10 |
| centuries to decades | centuries × 10 |
| centuries to years | centuries × 100 |
Fractions of a unit
Worked examples
12 hour = (12 × 60) minutes = 30 minutes.1 34 hours = (74 × 60) minutes = 105 minutes.
13 day = (13 × 24) hours = 8 hours.
5 18 days = 5 × 24 + 18 × 24 = 120 + 3 = 123 hours.
9 16 years = 9 × 12 + 16 × 12 = 108 + 2 = 110 months.
1 12 decades = (32 × 10) years = 15 years.
25 century = (25 × 10) decades = 4 decades. 910 century = (910 × 100) years = 90 years.
The strip below shows 1 century as 10 decades of 10 years each. You can see why 15 century is 20 years and 910 century is 90 years.
Decimals
Worked examples
4.5 hours = (4.5 × 60) minutes = 270 minutes. Or: 4 hours + 0.5 hour = 240 + 30 = 270 minutes.0.25 day = (0.25 × 24) hours = 6 hours. 1.75 days = 42 hours.
29.5 years = (29.5 × 12) months = 354 months.
1.3 decades = 13 years (move the decimal point one place to the right).
5.1 centuries = (5.1 × 10) decades = 51 decades. 0.67 century = 67 years.
0.35 hour is not 35 minutes. An hour has 60 minutes, not 100: 0.35 × 60 = 21 minutes. Likewise, 1.5 hours is 1 hour 30 minutes, not "1 hour 50 minutes".
Go the other way by dividing. 135 minutes ÷ 60 = 2 remainder 15, so 135 minutes = 2 hours 15 minutes. Mental maths helps: 0.5 day = 12 hours, 0.25 day = 6 hours, 0.5 year = 6 months.
Convert (a) 56 hour to minutes and (b) 6.25 years to months.
Show answerHide answer
(a) 56 × 60 = 50 minutes. (b) 6.25 × 12 = 6 × 12 + 0.25 × 12 = 72 + 3 = 75 months.
4.3Adding and subtracting time
Time can be given in fractions, in decimals or in two units together (such as 2 days 9 hours). Use the same ideas you know from fractions and decimals.
Adding
- Fractions of the same unit: make the denominators the same first, then add.
- Decimals of the same unit: line up the decimal points, then add as for whole numbers.
- Two units: add each unit separately. If the smaller unit reaches its limit (60 minutes, 24 hours, 12 months, 10 years), regroup it into one of the bigger unit.
Worked examples
13 hour + 1 56 hours = 26 hour + 1 56 hours = 1 76 hours = 2 16 hours.1.5 hours + 0.75 hour = 2.25 hours.
4.75 days + 9 days 18 hours: 4.75 days = 4 days 18 hours (0.75 × 24 = 18). Days: 4 + 9 = 13. Hours: 18 + 18 = 36 = 1 day 12 hours. Total = 14 days 12 hours.
2 12 years + 5 years 9 months = 2 years 6 months + 5 years 9 months = 7 years 15 months = 8 years 3 months.
Do not leave "7 years 15 months" as the final answer: 12 months make 1 year, so regroup it as 8 years 3 months. Always check that the smaller unit is below its limit (under 60 minutes, under 24 hours, under 12 months).
Subtracting
- Convert to the same unit first. For example, change hours to minutes.
- If you cannot take away, borrow one of the bigger unit: 1 day = 24 hours, 1 year = 12 months, 1 century = 10 decades.
Worked examples
1 56 hours − 12 minutes: 56 hour = 50 minutes, so 1 hour 50 minutes − 12 minutes = 1 hour 38 minutes.3.45 hours − 1 hour 40 minutes: 3.45 hours = 3 hours 27 minutes. 3 hours 27 minutes − 1 hour 40 minutes = 2 hours 87 minutes − 1 hour 40 minutes = 1 hour 47 minutes.
9.75 days − 3 days 5 hours: 9 days 18 hours − 3 days 5 hours = 6 days 13 hours.
14.5 years − 7 34 years: 14.5 − 7.75 = 6.75 years = 6 years 9 months.
0.23 century − 0.06 century = 0.17 century.
14.2 centuries − 3 centuries 9 decades: 14 centuries 2 decades = 13 centuries 12 decades. So 13 centuries 12 decades − 3 centuries 9 decades = 10 centuries 3 decades.
Check subtraction by adding back. 6 days 13 hours + 3 days 5 hours = 9 days 18 hours = 9.75 days. Correct.
(a) 3 days 9 hours + 34 day. (b) 5 years − 2 years 8 months.
Show answerHide answer
(a) 34 day = 18 hours. 3 d 9 h + 18 h = 3 d 27 h = 4 days 3 hours. (b) 4 years 12 months − 2 years 8 months = 2 years 4 months.
4.4Solving problems about time
Use four steps: understand the problem, plan a strategy, solve, then check. Useful strategies are drawing a diagram, trial and error, and working backwards.
Shelf life of a food product
A packet of curry paste was made on 9.12.2019 and expires on 8.2.2021. Find its shelf life in days.Step 1. 9.12.2019 to 31.12.2019 = 31 − 9 + 1 = 23 days.
Step 2. The whole of 2020 (leap year) = 366 days.
Step 3. 1.1.2021 to 31.1.2021 = 31 days. 1.2.2021 to 8.2.2021 = 8 days.
Step 4. Total = 23 + 366 + 31 + 8 = 428 days. Check by counting on a calendar.
Work backwards: two antique vases
Together two vases are 318 years old. One vase is 1.15 centuries old. How old is the other?Step 1. 1.15 centuries = 1.15 × 100 = 115 years.
Step 2. 318 − 115 = 203 years.
Step 3. 203 years = 2 centuries 3 years. Check: 2 centuries 3 years + 1.15 centuries = 203 + 115 = 318 years.
Trial and error: when is one building 3 times the age of another?
In 2020 a building is 0.4 decade old and another is 2 decades old. Try each year and multiply the younger age by 3.2021: 0.5 × 3 = 1.5 decades, but the other is 2.1. Too small.
2022: 0.6 × 3 = 1.8, other 2.2. 2023: 0.7 × 3 = 2.1, other 2.3.
2024: 0.8 × 3 = 2.4 and the other is 2.4. Equal, so the answer is the year 2024.
The shelf life of a snack is from 17.02.2020 to 17.08.2020 (leap year). How many days is that? (Count both the first and last day.)
Show answerHide answer
17 to 29 February = 13 days. March 31, April 30, May 31, June 30, July 31 = 153 days. 1 to 17 August = 17 days. Total = 13 + 153 + 17 = 183 days.
Summary
- Duration = end − start. Count the first day too when you count days between dates (+ 1).
- 31 days: Jan, Mar, May, Jul, Aug, Oct, Dec. 30 days: Apr, Jun, Sep, Nov. February has 28 days, or 29 in a leap year (year ÷ 4 with no remainder).
- 1 hour = 60 minutes; 1 day = 24 hours; 1 year = 12 months; 1 decade = 10 years; 1 century = 10 decades = 100 years.
- Bigger to smaller unit: multiply, for fractions and decimals alike. 0.35 hour = 21 minutes.
- Add and subtract fractions with the same denominator and decimals with the points lined up; for two units, regroup.
- Solve problems in four steps: understand, plan, solve, check.