Primary & Middle School Mathematics · Grades 1–9

12Money and Measures

Measuring rulers, kitchen scales, jugs, coins, clocks and calendars: this chapter is about the numbers of everyday life — how long, how heavy, how much it holds, how much it costs, and what time it is.

12.1 Lengths, masses, capacities

Definition 12.1 (Everyday units)

Length: centimeter (cm) for small things, meter (m) for big ones, with $1$ m $= 100$ cm;
mass: gram (g) for light things, kilogram (kg) for heavy ones, with $1$ kg $= 1000$ g;
capacity (how much a container holds): liter (L).

Smaller units and their exact rules come in Chapter 19.

Example 12.2 (Which unit?)

A crayon: about $9$ cm. A door: about $2$ m. An apple: about $150$ g. A big bag of flour: $1$ kg. A milk bottle: $1$ L. Guess the unit before measuring — it stops silly answers like “the door is $2$ cm”.

12.2 Money

Method 12.3 (Making an amount)

With coins and bills of $1$ , $2$ , $5$ , $10$ , $20$ :

start with the biggest that fits, then fill the rest;
to make $27$ : a $20$ , a $5$ , and a $2$ ;
there are often several ways — $27$ is also $10 + 10 + 5 + 2$ .

Making 27 with a bill and two coins — and there are other ways, like 10 + 10 + 5 + 2. — Making $27$ with a bill and two coins — and there are other ways, like $10 + 10 + 5 + 2$ .

Example 12.4 (Change)

A book costs $14$ ; you pay with a $20$ . The change is $20 - 14 = 6$ , found by counting up: $14 \to 20$ is $6$ (Method 9.4).

12.3 Time

Method 12.5 (Half and quarter hours)

The long hand tells the minutes past the hour:

straight up ( $12$ ): on the hour — $3{:}00$ ;
quarter turn to the right ( $3$ ): quarter past — $3{:}15$ ;
straight down ( $6$ ): half past — $3{:}30$ ;
three-quarter turn ( $9$ ): quarter to the next hour — $3{:}45$ , “quarter to four”.

Quarter past, half past, quarter to: the long red hand at the 3, the 6, the 9. — Quarter past, half past, quarter to: the long red hand at the $3$ , the $6$ , the $9$ .

Example 12.6 (The calendar)

A year has $12$ months, in order: January, February, March, April, May, June, July, August, September, October, November, December. Most months have $30$ or $31$ days; February has $28$ (or $29$ ). A date names the day and the month: “the $5$ th of March”.

12.4 Exercises

Exercise 12.1 ★

Choose the unit (cm, m, g, kg or L): a pencil; the classroom length; a feather; a schoolbag; a water bottle.

Solution

Solution of Exercise 12.1.

Pencil: cm. Classroom length: m. Feather: g. Schoolbag: kg. Water bottle: L.

Exercise 12.2 ★

Convert: $2$ m into cm; $300$ cm into m; $1$ kg into g.

Solution

Solution of Exercise 12.2.

$2$ m $= 200$ cm; $300$ cm $= 3$ m; $1$ kg $= 1000$ g.

Exercise 12.3 ★

Make these amounts with coins and bills: $16$ ; $23$ ; $35$ . Give one way for each.

Solution

Solution of Exercise 12.3.

For example: $16 = 10 + 5 + 1$ ; $23 = 20 + 2 + 1$ ; $35 = 20 + 10 + 5$ . (Other ways are possible.)

Exercise 12.4 ★

Compute the change: pay $10$ for a $6$ toy; pay $20$ for a $17$ book; pay $50$ for a $32$ game.

Solution

Solution of Exercise 12.4.

$10 - 6 = 4$ ; $20 - 17 = 3$ ; $50 - 32 = 18$ .

Exercise 12.5 ★

Read the time: long hand on $6$ , short hand between $4$ and $5$ ; long hand on $3$ , short hand just after $8$ .

Solution

Solution of Exercise 12.5.

Long hand on $6$ , short between $4$ and $5$ : half past four ( $4{:}30$ ). Long hand on $3$ , short just after $8$ : quarter past eight ( $8{:}15$ ).

Exercise 12.6 ★

Draw a clock showing half past two. Where does each hand point?

Solution

Solution of Exercise 12.6.

Half past two: the long hand points down to the $6$ , the short hand sits between the $2$ and the $3$ .

Exercise 12.7 ★

Name the month: after May; before January; the month of your birthday.

Solution

Solution of Exercise 12.7.

After May: June. Before January: December. (Birthday month varies.)

Exercise 12.8 ★

Lea buys a drink for $3$ and a cake for $4$ . She pays with a $10$ bill. How much change?

Solution

Solution of Exercise 12.8.

Spent: $3 + 4 = 7$ . Change: $10 - 7 = 3$ .

Exercise 12.9 ★

Which is heavier: a $1$ kg bag of sugar or $800$ g of apples? By how many grams?

Solution

Solution of Exercise 12.9.

$1$ kg $= 1000$ g, which is more than $800$ g: the sugar is heavier, by $1000 - 800 = 200$ g.

Exercise 12.10 ★★

Make $40$ with exactly four coins or bills (chosen from $1$ , $2$ , $5$ , $10$ , $20$ ). Find two different ways.

Solution

Solution of Exercise 12.10.

For example $20 + 10 + 5 + 5 = 40$ and $10 + 10 + 10 + 10 = 40$ (two ways with exactly four coins or bills).