Generate fractions and decimals between two quantities

Ontario Grade 7 Mathematics

Number

Ontario Grade 7 Mathematics — B1.5

A number can fit between two other numbers, even when the numbers look close together. For example, decimal numbers fit between 0.4 and 0.5. Fractions also fit between one third and one half. In this lesson, you will find numbers between given quantities and check your choices. A number is between two quantities when it is greater than the smaller quantity and less than the larger quantity.

What you will learn

Explain what it means for a fraction or decimal to be between two quantities.
Generate a fraction between two fractions by rewriting them with equal-sized parts.
Generate a decimal between two quantities by choosing a suitable place value.
Check that a proposed number is greater than the smaller quantity and less than the larger one.

1. Understand what “between” means

A number line shows numbers in order. As you move to the right, the numbers get larger. If a number is between two others, it sits to the right of the smaller number and to the left of the larger number. For example, 0.6 is between 0.5 and 0.7 because 0.5 is less than 0.6, and 0.6 is less than 0.7.

A fraction names equal parts of a whole. In the fraction three eighths, the denominator, 8, tells how many equal parts make one whole. The numerator, 3, tells how many of those parts are counted. Fractions and decimals can name the same amount, so either form can help us find a number between two quantities.

Start by identifying the smaller quantity and the larger quantity. Then choose a value that is greater than the smaller one and less than the larger one. The chosen value does not have to be exactly halfway between them. More than one answer may work.

a<x<b

A number between two quantities is greater than the smaller and less than the larger.
A number line shows this order from left to right.
Fractions and decimals can name the same quantity.

2. Generate a fraction between two fractions

Fractions are easier to compare when they name equal-sized parts. Fractions with the same denominator have a common denominator. For example, fifths and tenths are different-sized parts, but one fifth is the same amount as two tenths. This lets us rewrite fifths as tenths.

Suppose the quantities are two fifths and three fifths. There is no whole number of fifths strictly between them. We can split each fifth into two equal parts. Then two fifths is the same as four tenths, and three fifths is the same as six tenths. Five tenths is between them.

An equivalent fraction looks different but has the same value as the original fraction. To make an equivalent fraction, multiply the numerator and denominator by the same number. For example, multiplying both by 2 changes two fifths into four tenths without changing its value.

For fractions with different denominators, rewrite them as equivalent fractions with a shared denominator. Then find a numerator that gives a fraction between the rewritten quantities. Once the denominators match, comparing the numerators compares the fractions. Check that your answer is strictly between the endpoints, not equal to either one.

\frac{2}{5}=\frac{4}{10}<\frac{5}{10}<\frac{6}{10}=\frac{3}{5}

A common denominator gives fractions equal-sized parts.
Equivalent fractions have the same value.
With equal denominators, a numerator between the endpoint numerators gives a fraction between them.

3. Generate a decimal between two quantities

Decimals use place value. The first digit to the right of the decimal point counts tenths. The second digit counts hundredths, and the third counts thousandths. For example, 0.3 means three tenths.

A zero added at the end of a decimal does not change its value. So 0.4 and 0.40 name the same amount. Adding a zero can help you line up decimal places and compare quantities.

To find a decimal between two quantities, write them with the same number of decimal places. Then choose a value that is greater than the smaller quantity and less than the larger one. If the quantities are close, use another decimal place to look for a value in the gap.

For example, rewrite 0.4 as 0.40. The decimal 0.41 is greater than 0.40 and less than 0.50. You can check the comparisons by lining up the digits. Do not decide only by counting digits: compare the values.

0.40<0.41<0.50

Adding zeros at the end of a decimal does not change its value.
Matching decimal places can make comparisons easier.
A smaller place value can help you find a number in a narrow gap.

4. Choose and check an answer

A number line can help you picture the gap. Equivalent fractions help when the quantities are fractions. Matching decimal places helps when the quantities are decimals. Choose a method that suits the values in the question.

Do not choose an endpoint. If you need a number between one quarter and one half, one quarter and one half are not answers. A valid answer must be greater than one quarter and less than one half.

For practice, find a fraction between one sixth and one third, and a decimal between 0.72 and 0.73. Check each choice against both endpoints. The worked examples show how to make and check these choices.

Endpoints are not between themselves.
Check both comparisons before accepting an answer.
Many different fractions or decimals may fit in the same gap.

Worked example

A fraction between fifths

Generate a fraction between two fifths and three fifths.

Tenths between two fifths

The labelled values show that five tenths lies between four tenths and six tenths.

Rewrite as tenths
The fractions have the same denominator. Split each fifth into two equal parts so both quantities can be written in tenths.
$\frac{2}{5}=\frac{4}{10},\quad \frac{3}{5}=\frac{6}{10}$
Choose and check
Five tenths is greater than four tenths and less than six tenths. It is therefore between the original fractions.
$\frac{4}{10}<\frac{5}{10}<\frac{6}{10}$

Answer:

\frac{5}{10}

, which is also

\frac{1}{2}

.

Check: Two fifths is four tenths, and three fifths is six tenths. Five tenths is strictly between these values.

Worked example

A decimal in a small gap

Generate a decimal between 0.72 and 0.73.

A thousandth between hundredths

The selected value 0.725 lies between 0.720 and 0.730.

Write equal decimal places
The endpoints are hundredths. Add a zero at the end of each decimal so they are written in thousandths. The values stay the same.
$0.72=0.720,\quad 0.73=0.730$
Choose a thousandth
The decimal 0.725 is greater than 0.720 and less than 0.730. It fits strictly between the given quantities.
$0.720<0.725<0.730$

Answer:

0.725

Check: The value 0.725 is five thousandths greater than 0.720 and five thousandths less than 0.730. It is not an endpoint.

Worked example

Fractions with different denominators

Generate a fraction between one sixth and one third.

Twelfths between sixths and thirds

Two twelfths, three twelfths, and four twelfths are shown in increasing order.

Rewrite as twelfths
Use twelfths so both fractions name equal-sized parts. One sixth is two twelfths, and one third is four twelfths. The numerators are two and four.
$\frac{1}{6}=\frac{2}{12},\quad \frac{1}{3}=\frac{4}{12}$
Choose a fraction between
Three twelfths is greater than two twelfths and less than four twelfths. Therefore, it is between the original quantities.
$\frac{2}{12}<\frac{3}{12}<\frac{4}{12}$

Answer:

\frac{3}{12}

, which is also

\frac{1}{4}

.

Check: The endpoint fractions are two twelfths and four twelfths. Three twelfths is strictly between them.

Common mistakes and how to avoid them

Choosing one of the two endpoints as the answer.

Correction: A number between the quantities must be greater than the smaller endpoint and less than the larger endpoint.

Comparing fraction numerators when the denominators are different.

Correction: First rewrite the fractions with a common denominator. Then compare the numerators.

Thinking that adding a zero at the end of a decimal changes its value.

Correction: A zero added at the end does not change the value. For example, 0.7 and 0.70 name the same amount.

Giving a value without checking both sides.

Correction: Check that the proposed value is greater than the smaller quantity and less than the larger one.

Lesson summary

To generate a number between two quantities, choose one greater than the smaller quantity and less than the larger quantity.
For fractions, equivalent fractions with a common denominator make equal-sized parts easy to compare.
For decimals, match decimal places and use a smaller place value when the gap is narrow.
Always check that your answer is not an endpoint.

Check your understanding

Question 1

Which fraction is between three eighths and five eighths?

$\frac{2}{8}$
$\frac{4}{8}$
$\frac{6}{8}$
correctIndex

Show answer and explanation

\frac{4}{8}

Four eighths is greater than three eighths and less than five eighths.

Question 2

Which decimal is between 0.6 and 0.7?

0.59
0.65
0.70
correctIndex

Show answer and explanation

0.65

0.65 is greater than 0.6 and less than 0.7. The first choice is smaller than the lower endpoint, and the last is an endpoint.

Question 3

Which fraction is between one quarter and one half?

$\frac{1}{8}$
$\frac{3}{8}$
$\frac{5}{8}$
correctIndex

Show answer and explanation

\frac{3}{8}

One quarter is two eighths, and one half is four eighths. Three eighths is between them.

Key terms

Between: Greater than one quantity and less than the other.
Denominator: The bottom number in a fraction. It tells how many equal parts make one whole.
Numerator: The top number in a fraction. It tells how many parts are counted.
Equivalent fractions: Fractions that look different but name the same amount.
Common denominator: A shared denominator that lets fractions be written with equal-sized parts.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 7 Mathematics, expectation B1.5. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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B1.5 · Generate fractions and decimals between two quantities

Ontario Grade 7 Mathematics

Ontario Grade 7 Mathematics — B1.5

What you will learn

1. Understand what “between” means

2. Generate a fraction between two fractions

3. Generate a decimal between two quantities

4. Choose and check an answer

A fraction between fifths

A decimal in a small gap

Fractions with different denominators

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through Grade 7 Math

About this lesson and its review