Primary & Middle School Mathematics · Grades 1–9

19Measuring

How long, how heavy, how much does it hold, how late is it? Measuring turns the world into numbers — provided everyone uses the same units. This chapter presents the units of length, mass and capacity, and teaches how to read a clock.

19.1 Lengths

Definition 19.1 (Units of length)

The basic unit of length is the meter (m). For small lengths: the centimeter (cm) and the millimeter (mm); for long distances: the kilometer (km).

1 \text{ m} = 100 \text{ cm}, \qquad 1 \text{ cm} = 10 \text{ mm}, \qquad 1 \text{ km} = 1\,000 \text{ m}.

Reading a ruler: big marks for centimeters, small marks for millimeters. The object starts at the 0 mark — not at the edge of the ruler! — and ends at the 6th small mark after the 3. — Reading a ruler: big marks for centimeters, small marks for millimeters. The object starts at the $0$ mark — not at the edge of the ruler! — and ends at the $6$ th small mark after the $3$ .

Example 19.2 (Choosing the right unit)

An ant: about $5$ mm. A pencil: about $18$ cm. A door: about $2$ m. A walk to the next village: about $3$ km. Giving “the length of a pencil is $18$ ” means nothing — the unit is part of the answer.

Method 19.3 (Converting units)

To convert, multiply or divide by $10$ , $100$ or $1\,000$ :

from a big unit to a small one, multiply: $3$ m $= 3 \times 100 = 300$ cm;
from a small unit to a big one, divide: $250$ cm $= 250 \div 100 = 2$ m and $50$ cm;
to add lengths, convert them to the same unit first: $1$ m $+ 35$ cm $= 100 + 35 = 135$ cm.

19.2 Masses and capacities

Definition 19.4 (Units of mass and capacity)

Masses are measured in grams (g) and kilograms (kg), capacities (how much a container holds) in liters (L) and centiliters (cL):

1 \text{ kg} = 1\,000 \text{ g}, \qquad 1 \text{ L} = 100 \text{ cL}.

Example 19.5

A letter: about $20$ g. A bag of flour: $1$ kg. A bathtub: about $150$ L. A glass: about $20$ cL. To balance a scale holding a $1$ kg bag against $650$ g of apples, add $1\,000 - 650 = 350$ g on the apples’ side.

19.3 Reading the clock

Method 19.6 (Reading the time)

On a clock with hands:

the short hand gives the hour: read the number it has passed last;
the long hand gives the minutes: each big mark is $5$ minutes, so a long hand on the $3$ means $15$ minutes;
after half past, one can also read “to the next hour”: $10{:}50$ is “ten to eleven”.

Three clocks: short blue hand for hours, long red hand for minutes. At 7:30 the hour hand sits between 7 and 8; at 10:50 it is almost on the 11. — Three clocks: short blue hand for hours, long red hand for minutes. At 7:30 the hour hand sits *between* 7 and 8; at 10:50 it is almost on the 11.

Example 19.7 (Durations)

The film starts at $14{:}40$ and ends at $16{:}10$ . How long is it? Count in two hops:

from $14{:}40$ to $15{:}00$ : $20$ minutes;
from $15{:}00$ to $16{:}10$ : $1$ hour $10$ minutes.

Total: $1$ h $30$ min. Careful: an hour has $60$ minutes, so $16{:}10$ minus $14{:}40$ cannot be computed like $1610 - 1440$ !

19.4 Exercises

Exercise 19.1 ★

Which unit would you use (mm, cm, m or km) for: the width of a nail; the height of a classmate; the length of the schoolyard; the distance between two towns?

Solution

Solution of Exercise 19.1.

Nail width: mm. Classmate’s height: cm (or m). Schoolyard: m. Distance between towns: km.

Exercise 19.2 ★

Convert: $5$ m into cm; $70$ mm into cm; $4$ km into m; $600$ cm into m.

Solution

Solution of Exercise 19.2.

$5$ m $= 500$ cm; $70$ mm $= 7$ cm; $4$ km $= 4\,000$ m; $600$ cm $= 6$ m.

Exercise 19.3 ★

Measure with your ruler (to the nearest millimeter) the sides of a triangle drawn by a classmate, and compute the total length of its border.

Solution

Solution of Exercise 19.3.

Answers depend on the triangle; the total length of the border is the sum of the three measured sides (this total is called the perimeter, studied in Chapter 28).

Exercise 19.4 ★

Compute, converting first: $2$ m $+ 45$ cm (in cm); $1$ km $+ 250$ m (in m); $3$ cm $+ 8$ mm (in mm).

Solution

Solution of Exercise 19.4.

$2$ m $+ 45$ cm $= 200 + 45 = 245$ cm.

$1$ km $+ 250$ m $= 1\,000 + 250 = 1\,250$ m.

$3$ cm $+ 8$ mm $= 30 + 8 = 38$ mm.

Exercise 19.5 ★

Convert: $3$ kg into g; $2\,500$ g into kg and g; $4$ L into cL; $350$ cL into L and cL.

Solution

Solution of Exercise 19.5.

$3$ kg $= 3\,000$ g; $2\,500$ g $= 2$ kg and $500$ g; $4$ L $= 400$ cL; $350$ cL $= 3$ L and $50$ cL.

Exercise 19.6 ★

A recipe needs $750$ g of flour. Mona has a $1$ kg bag. Does she have enough? How much will be left?

Solution

Solution of Exercise 19.6.

$1$ kg $= 1\,000$ g $> 750$ g: yes, she has enough, and $1\,000 - 750 = 250$ g will be left.

Exercise 19.7 ★

Read the times: the short hand is between $2$ and $3$ , the long hand on the $6$ ; the short hand on the $9$ , the long hand on the $12$ ; the short hand almost on the $5$ , the long hand on the $10$ .

Solution

Solution of Exercise 19.7.

Short hand between $2$ and $3$ , long hand on the $6$ : $2{:}30$ . Short hand on $9$ , long hand on $12$ : $9{:}00$ . Short hand almost on $5$ , long hand on $10$ : $4{:}50$ (“ten to five” — the hour hand has not reached the $5$ yet).

Exercise 19.8 ★

School starts at $8{:}30$ and the morning ends at $11{:}45$ . How long is the morning? (Count in hops, as in Example 19.7.)

Solution

Solution of Exercise 19.8.

From $8{:}30$ to $9{:}00$ : $30$ min; from $9{:}00$ to $11{:}45$ : $2$ h $45$ min. Total: $3$ h $15$ min.

Exercise 19.9 ★

The train leaves at $9{:}50$ and the ride lasts $45$ minutes. At what time does it arrive? (Hop to $10{:}00$ first.)

Solution

Solution of Exercise 19.9.

From $9{:}50$ to $10{:}00$ : $10$ min; there remain $45 - 10 = 35$ min after $10{:}00$ . Arrival: $10{:}35$ .

Exercise 19.10 ★★

A bottle holds $1$ L. Paul fills glasses of $20$ cL. How many full glasses can he pour? (Convert first.)

Solution

Solution of Exercise 19.10.

$1$ L $= 100$ cL, and $100 = 20 \times 5$ : Paul can pour $5$ full glasses.

Exercise 19.11 ★★

Lena walks $600$ m to school, and back, every day. How many meters does she walk in a $5$ -day school week? Give the answer in meters, then in kilometers.

Solution

Solution of Exercise 19.11.

Each day: $600 + 600 = 1\,200$ m. In five days: $1\,200 \times 5 = 6\,000$ m $= 6$ km.