Learn on PengiThe Art of Problem Solving: Prealgebra (AMC 8)Chapter 6: Decimals

Lesson 3: Decimals and Fractions

In this Grade 4 AMC 8 lesson from The Art of Problem Solving: Prealgebra, students learn how to convert between decimals and fractions by expressing decimals as powers of 10 and by rewriting fraction denominators as powers of 10. The lesson covers converting terminating decimals such as 0.125, -1.72, and 2.5625 into simplified fractions, as well as converting fractions like 7/8 and 19/32 into decimal form. Students also practice simplifying results using prime factorization and finding reciprocals of decimals.

Section 1

Convert decimals to mixed numbers

Property

To convert a decimal to a fraction or mixed number:

The number to the left of the decimal is the whole number part of the mixed number. If it is zero, you will have a proper fraction.
The number to the right of the decimal point becomes the numerator of the fraction.
The denominator is a power of 10 corresponding to the place value of the last digit.

Examples

To convert 4.09, the whole number is 4. The numerator is 9 and the place value is hundredths, so it is $4\frac{9}{100}$ .
To convert 3.7, the whole number is 3. The numerator is 7 and the place value is tenths, resulting in $3\frac{7}{10}$ .
The decimal $-0.286$ has no whole number part. The fraction is $-\frac{286}{1000}$ , which simplifies to $-\frac{143}{500}$ .

Explanation

Converting a decimal is straightforward. The whole number stays the same. The digits after the decimal point go on top of the fraction, and the bottom is the place value of the very last digit (like 10, 100, or 1000).

Section 2

Convert fractions to decimals

Property

To convert a fraction to a decimal, divide the numerator of the fraction by the denominator of the fraction. The fraction bar indicates division, so a fraction like $\frac{a}{b}$ can be written as $a \div b$ .

Examples

To write $\frac{3}{4}$ as a decimal, we divide 3 by 4. The calculation $3.00 \div 4$ gives us $0.75$ . So, $\frac{3}{4} = 0.75$ .

To convert the improper fraction $\frac{9}{5}$ to a decimal, we divide 9 by 5. The calculation $9.0 \div 5$ results in $1.8$ . So, $\frac{9}{5} = 1.8$ .

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Convert decimals to mixed numbers

Property

To convert a decimal to a fraction or mixed number:

The number to the left of the decimal is the whole number part of the mixed number. If it is zero, you will have a proper fraction.
The number to the right of the decimal point becomes the numerator of the fraction.
The denominator is a power of 10 corresponding to the place value of the last digit.

Examples

To convert 4.09, the whole number is 4. The numerator is 9 and the place value is hundredths, so it is $4\frac{9}{100}$ .
To convert 3.7, the whole number is 3. The numerator is 7 and the place value is tenths, resulting in $3\frac{7}{10}$ .
The decimal $-0.286$ has no whole number part. The fraction is $-\frac{286}{1000}$ , which simplifies to $-\frac{143}{500}$ .

Explanation

Section 2

Convert fractions to decimals

Property

Examples

To write $\frac{3}{4}$ as a decimal, we divide 3 by 4. The calculation $3.00 \div 4$ gives us $0.75$ . So, $\frac{3}{4} = 0.75$ .

To convert the improper fraction $\frac{9}{5}$ to a decimal, we divide 9 by 5. The calculation $9.0 \div 5$ results in $1.8$ . So, $\frac{9}{5} = 1.8$ .

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Convert decimals to mixed numbers

Property

To convert a decimal to a fraction or mixed number:

The number to the left of the decimal is the whole number part of the mixed number. If it is zero, you will have a proper fraction.
The number to the right of the decimal point becomes the numerator of the fraction.
The denominator is a power of 10 corresponding to the place value of the last digit.

Examples

To convert 4.09, the whole number is 4. The numerator is 9 and the place value is hundredths, so it is $4\frac{9}{100}$ .
To convert 3.7, the whole number is 3. The numerator is 7 and the place value is tenths, resulting in $3\frac{7}{10}$ .
The decimal $-0.286$ has no whole number part. The fraction is $-\frac{286}{1000}$ , which simplifies to $-\frac{143}{500}$ .

Explanation

Section 2

Convert fractions to decimals

Property

Examples

To write $\frac{3}{4}$ as a decimal, we divide 3 by 4. The calculation $3.00 \div 4$ gives us $0.75$ . So, $\frac{3}{4} = 0.75$ .

To convert the improper fraction $\frac{9}{5}$ to a decimal, we divide 9 by 5. The calculation $9.0 \div 5$ results in $1.8$ . So, $\frac{9}{5} = 1.8$ .