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Expanding, simplifying and comparing decimal numbers

Expanding and simplifying decimal numbers

Example 1

Zoe bought \latex{ 0.9 } \latex{ kg }, Nora bought \latex{ 0.90 } \latex{ kg }, and Tim bought \latex{ 0.900 } \latex{ kg } of strawberries. Who bought the most?

Solution

Write the values in fraction form, then find a common denominator.

Zoe: \latex{0.9\;kg=\frac{9}{10}\;kg=\frac{900}{1000}\;kg};

Nora: \latex{0.90\;kg=\frac{90}{100}\;kg=\frac{900}{1000}\;kg};

Tim: \latex{0.900\;kg=\frac{900}{1000}\;kg}.

All three of them bought the same amount of strawberries, so:

\latex{0.9\;kg = 0.90\;kg = 0.900\;kg.}

Expanding a decimal number means writing zeros after the last digit.

\latex{0.9=0.90=0.900=0.9000=0.90000=0.900000=0.9000000=...}

Expanding

Simplifying

Simplifying a decimal number means eliminating the zeros at the end of the number.

The value of the decimal number does not change during expansion and simplification.

Comparing decimal numbers

Example 2

In a long jump competition, Pete's result was \latex{ 4.28 } \latex{ metres }, while Paul's was \latex{ 4.3 } \latex{ metres }. Who jumped farther?

Solution

a) Compare them by converting the units.

Pete: \latex{ 4.28 } \latex{ m = 4 } \latex{ m } \latex{ 28 } \latex{ cm = } \latex{ 428 } \latex{ cm };
Paul: \latex{ 4.3 } \latex{ m } \latex{ = 4 } \latex{ m } \latex{ 3 } \latex{ dm = } \latex{ 430 } \latex{ cm }.
Since \latex{ 428\lt 430 }, Paul jumped farther than Pete.

b) Compare them without converting the units.

Pete jumped \latex{4.28=4\frac{28}{100}\;m}, Paul \latex{4.3=4\frac{3}{10}=4\frac{30}{100}=4.30\;m}.

\latex{4.28=4\frac{28}{100}\lt 4\frac{30}{100}=4.30=4.3}. So Paul jumped further.

Example 3

Compare the following decimal numbers.

a) \latex{ 234.59 } and \latex{ 242.59 }

b) \latex{ 346.24 } and \latex{ 346.23 }

c) \latex{ 2.48 } and \latex{ 2.476 }

d) \latex{ 637.198 } and \latex{ 637.2 }

Solution

When comparing two decimal numbers, going from left to right, you must find the first place where the digits differ. By comparing these digits, you can decide which decimal number is greater.

a) \latex{2\fbox{3}4.59\lt2\fbox{4}2.59}

b) \latex{346.2\fbox{4}\gt346.2\fbox{3}}

c) \latex{2.4\fbox{8}\gt2.4\fbox{7}6}

d) \latex{637.\fbox{1}98\lt637.\fbox{2}}

Exercises

{{exercise_number}}. Write down three expanded forms of each of the following decimal numbers.

a) \latex{ 3.5 }

b) \latex{ 100.970 }

c) \latex{ 4.05 }

d) \latex{ 6.0 }

e) \latex{ 27.33 }

f) \latex{ 19 }

{{exercise_number}}. Simplify the decimal numbers (if possible).

a) \latex{ 4.50 }

b) \latex{ 6.200 }

c) \latex{ 72.030 }

d) \latex{ 808.000 }

e) \latex{ 10.00970 }

f) \latex{ 90.0 }

g) \latex{ 23.500 }

h) \latex{ 10.0020 }

i) \latex{ 60.00 }

j) \latex{ 9.000 }

k) \latex{ 5.0100 }

l) \latex{ 12.00006 }

{{exercise_number}}. The second hand on a clock is \latex{ 6.7 } \latex{ cm }, the minute hand is \latex{ 8.2 } \latex{ cm }, and the hour hand is \latex{ 5.5 } \latex{ cm } long. Arrange the hands in ascending order according to their length.

{{exercise_number}}. Which one of the two numbers is greater?

a) \latex{ 22.888 } or \latex{ 22.9 }

b) \latex{ 0.6400 } or \latex{ 0.639 }

c) \latex{ 18.986 } or \latex{ 19 }

d) \latex{ 7.0012 } or \latex{ 7.00099 }

e) \latex{ 75.8 } or \latex{ 78.8000 }

f) \latex{ 5457.3872 } or \latex{ 5459.12 }

{{exercise_number}}. Write three decimal numbers in place of each symbol if

a) \latex{2\lt\textcolor{#2d8113}{\blacksquare}\lt3};

b) \latex{1.2\lt\textcolor{#e3004f}{\blacktriangle}\lt1.3};

c) \latex{0.12\lt\textcolor{#009ee0}{\bullet}\lt0.13}.

{{exercise_number}}. The average length of a few insects:

common fruit fly: \latex{ 2.1 } \latex{ mm }

cockchafer: \latex{ 2.6 } \latex{ cm }

seven-spotted ladybird: \latex{ 6.8 } \latex{ mm }

stag beetle: \latex{ 6 } \latex{ cm }

great silver water beetle: \latex{ 4.1 } \latex{ cm }

hornet: \latex{ 2.4 } \latex{ cm }

Arrange the insects in ascending order according to their average length.

{{exercise_number}}. You bought \latex{ 1.8 } \latex{ kg } of apples, \latex{ 0.25 } \latex{ kg } of ham, \latex{ 900 } \latex{ g } of tomatoes, \latex{ 1,225 } \latex{ g } of meat, and \latex{ 530 } \latex{ g } of carrots. Convert these values into \latex{ kilograms } and arrange them in descending order.

{{exercise_number}}. The following tables show the results of two swimming finals at the \latex{ 2004 } Paralympics in Athens. The times are expressed in seconds. Arrange them in the correct order.

a) women’s \latex{ 50 } \latex{ m } butterfly

Lane

Name

Nationality

Time

\latex{ 55.23 }

\latex{ 49.50 }

\latex{ 48.88 }

\latex{ 46.57 }

\latex{ 44.70 }

\latex{ 50.84 }

\latex{ 1:04.08 }

\latex{ 1:05.25 }

SIN

HUN

UKR

ESP

CZE

ITA

CAN

Theresa Goh Ruisi

Diána Zámbó

Katalin Engelhardt

Olena Akopjan

Teresa Perales

Bela Hlavackova

Dalia Dameno

Kaley McLean

\latex{ 1 }

\latex{ 2 }

\latex{ 3 }

\latex{ 4 }

\latex{ 5 }

\latex{ 6 }

\latex{ 7 }

\latex{ 8 }

b) men’s \latex{ 50 } \latex{ m } backstroke

Lane

Name

Nationality

Time

\latex{ 43.13 }

\latex{ 42.67 }

\latex{ 40.49 }

\latex{ 37.53 }

\latex{ 40.14 }

\latex{ 42.78 }

\latex{ 43.45 }

\latex{ 43.50 }

BRA

FRA

HUN

CHIN

HUN

POL

GBR

MAS

Francisco Avelino

Pascal Pinard

Zsolt Vereckei

Junquan He

Ervin Kovács

Krzysztof Sleczka

Anthony Stephens

Sidi Abdullah

\latex{ 1 }

\latex{ 2 }

\latex{ 3 }

\latex{ 4 }

\latex{ 5 }

\latex{ 6 }

\latex{ 7 }

\latex{ 8 }

{{exercise_number}}. Write down the smallest decimal number with three decimal digits greater than \latex{ 12.7 }.

{{exercise_number}}. Write down the largest possible decimal number smaller than \latex{ 12.7 }.

Quiz

How many different decimal numbers can be made using the cards \latex{ 0 } \latex{ 1 } \latex{ 2 } . ? (A number cannot start or end with . ; numbers greater than \latex{ 1 } cannot start with \latex{ 0 }.)

Mathematics 5.

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