---
title: "Measuring"
book: "Primary & Middle School Mathematics"
subject: math
language: en
chapter: 19
exercises: 11
source: https://one-course.com/books/math/1/en/chapter/19-measuring
---

# Chapter 19 — Measuring

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](https://one-course.com/books/math/1/en/chapter/12-money-and-measures#def-g2-measure-units) and [capacity](https://one-course.com/books/math/1/en/chapter/12-money-and-measures#def-g2-measure-units), 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.](https://one-course.com/images/onecourse/chapters/math-1/g3-measure/fig-88722a6ee284.svg)

*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 $1\,000$:

1. from a big unit to a small one, *multiply* : $3$ m $= 3 \times 100 = 300$ cm;
2. from a small unit to a big one, *divide* : $250$ cm $= 250 \div 100 = 2$ m and $50$ cm;
3. 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:

1. the *short* hand gives the hour: read the number it has passed last;
2. the *long* hand gives the minutes: each big mark is $5$ minutes, so a long hand on the $3$ means $15$ minutes;
3. 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.](https://one-course.com/images/onecourse/chapters/math-1/g3-measure/fig-04329e6d8c0e.svg)

*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:

1. from $14{:}40$ to $15{:}00$ : $20$ minutes;
2. 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 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 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 of Exercise 19.3.**

Answers depend on the triangle; the total length of the border is the [sum](https://one-course.com/books/math/1/en/chapter/2-addition-first-steps#def-g1-addition-def) of the three measured sides (this total is called the *perimeter*, studied in [Chapter 28](https://one-course.com/books/math/1/en/chapter/28-measures-and-perimeter#ch-g4-measure)).

**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 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 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 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 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](#ex-g3-measure-duration).)

**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 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 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 of Exercise 19.11.**

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