Decimals: Fresh Take

Lumen Learning

Decimals: Fresh Take

Write and name decimals
Turn a decimal into a fraction
Place decimals on a number line and order them
Solve equations using decimals

Write and Name Decimals

The Main Idea

Decimals represent fractions or parts of a whole, based on powers of ten, using a point known as a decimal point.

Writing and naming decimals involve expressing numbers that are less than one, or fractions, in decimal notation. A decimal is written with a decimal point, and the place value of each digit after the decimal point indicates its value – tenths, hundredths, thousandths, and so on. For instance, the decimal [latex]0.35[/latex] is pronounced “zero point three five” or “thirty-five hundredths”. Each digit in a decimal number has a specific place value and is used to represent fractional values.

Name each decimal:

[latex]4.3[/latex]
[latex]2.45[/latex]
[latex]0.009[/latex]
[latex]-15.571[/latex]

Write the following numbers as a decimal:

six and seventeen hundredths
fourteen and thirty-seven hundredths

In the following video, we show more examples of how to write the name of a decimal using a place value chart.

You can view the transcript for “Read and Write Decimals” here (opens in new window).

Write a Decimal as a Fraction

The Main Idea

Writing a decimal as a fraction requires understanding of the place value system. Each digit after the decimal point signifies a specific fractional part: tenths, hundredths, thousandths, etc.

To convert a decimal to a fraction, consider the place value of the last digit. For instance, [latex]0.25[/latex] means twenty-five hundredths and it is represented as the fraction [latex]\frac{25}{100}[/latex]. It is important to simplify the fraction to its lowest terms whenever possible, so [latex]\frac{25}{100}[/latex] simplifies to [latex]\frac{1}{4}[/latex].

Write each of the following decimal numbers as a fraction or a mixed number:

[latex]4.09[/latex]
[latex]3.7[/latex]
[latex]-0.286[/latex]

The following video shows more examples of writing decimals as fractions:

You can view the transcript for “Ex 1: Convert a Decimal to a Fraction” here (opens in new window).

Rounding Decimals

The Main Idea

Rounding decimals involves approximating a decimal to the nearest whole number, tenth, hundredth, or other decimal place value. The process is similar to rounding whole numbers. If the digit right after the place value you’re rounding to is [latex]5[/latex] or greater, you round up the last digit kept. If it’s less than [latex]5[/latex], you keep the digit as it is.

For example, if rounding [latex]3.78[/latex] to the nearest tenth, you look at the hundredths place ([latex]8[/latex]). Since [latex]8[/latex] is greater than [latex]5[/latex], you round up, and [latex]3.78[/latex] becomes [latex]3.8[/latex].

Round [latex]18.379[/latex] to the nearest:

tenth
whole number

Watch the following video to see an example of how to round a number to several different place values.

You can view the transcript for “Examples: Rounding Decimals” here (opens in new window).

Locating and Ordering Decimals With a Number Line

The Main Idea

Locating and ordering decimals on a number line involves understanding the value of decimals and their relative positions. First, identify the whole numbers that the decimal falls between. Then, partition the space between these whole numbers into tenths, hundredths, or thousandths, as needed. Plot the decimals on the line at the corresponding position.

For example, [latex]0.5[/latex] would fall halfway between [latex]0[/latex] and [latex]1[/latex]. When ordering decimals, start from the smallest (left on the number line) and move to the largest (right on the number line).

Locate [latex]-0.74[/latex] on a number line.

In the next video, we show more examples of how to locate a decimal on the number line.

You can view the transcript for “Example: Identify Decimals on the Number Line” here (opens in new window).

Order Decimals

The Main Idea

Ordering decimals involves arranging them from least to greatest or vice versa based on their numerical value. Begin by comparing the digits in the tenths place of each decimal. If the tenths place digits are the same, move to the hundredths place, then the thousandths place, and so on, until you find a difference. The decimal with the smaller digit in the first place where the decimals differ is the smaller decimal. If all places are the same, then the decimals are equal.

In the following video lesson we show how to order decimals using inequality notation by comparing place values, and by using fractions.

You can view the transcript for “Decimal Notation: Ordering Decimals” here (opens in new window).

Order the following decimals using [latex]<\text{ or }\text{>}[/latex]:

[latex]0.64[/latex] ____ [latex]0.6[/latex]
[latex]0.83[/latex] ____ [latex]0.803[/latex]

Use [latex]<\text{or}>[/latex] to order: [latex]-0.1[/latex] ____ [latex]- 0.8[/latex]

Determine Whether a Decimal is a Solution of an Equation

Determine whether each of the following is a solution of [latex]x - 0.7=1.5[/latex]

[latex]x=1[/latex]
[latex]x=-0.8[/latex]
[latex]x=2.2[/latex]

Solve Equations with Decimals

Solve: [latex]y+2.3=-4.7[/latex]

Solve: [latex]a - 4.75=-1.39[/latex]

In the following video, we show more examples of using the addition and subtraction properties of equality to solve linear equations that contain decimals.

You can view the transcript for “Solving One Step Equations Using Addition and Subtraction (Decimals)” here (opens in new window).

Solve: [latex]-4.8=0.8n[/latex]

Watch the next video to see how to solve another equation with decimals that requires division.

You can view the transcript for “Ex: Solve a One Step Equation With Decimals by Dividing” here (opens in new window).

Solve: [latex]{\Large\frac{p}{-1.8}}=-6.5[/latex]

The following video shows an example of how to solve an equation with decimals that requires multiplication.

You can view the transcript for “Ex: Solve a One Step Equation With Decimals by Multiplying” here (opens in new window).

Solving Equations By Clearing Decimals

Solve: [latex]0.06x+0.02=0.25x - 1.5[/latex]

The next example uses an equation that is typical of the ones we will see in the money applications. Notice that we will distribute the decimal first before we clear all decimals in the equation.

Solve: [latex]0.25x+0.05\left(x+3\right)=2.85[/latex]

In the following video we present another example of how to solve an equation that contains decimals and variable terms on both sides of the equal sign.

You can view the transcript for “Ex: Solve a Linear Equation With Decimals and Variables on Both Sides” here (opens in new window).

[latex]4.3[/latex]
Name the number to the left of the decimal point.	four_____
Write “and” for the decimal point.	four and_____
Name the number to the right of the decimal point as if it were a whole number.	four and three_____
Name the decimal place of the last digit.	four and three tenths

[latex]2.45[/latex]
Name the number to the left of the decimal point.	two_____
Write “and” for the decimal point.	two and_____
Name the number to the right of the decimal point as if it were a whole number.	two and forty-five_____
Name the decimal place of the last digit.	two and forty-five hundredths

[latex]0.009[/latex]
Name the number to the left of the decimal point.	Zero is the number to the left of the decimal; it is not included in the name.
Name the number to the right of the decimal point as if it were a whole number.	nine_____
Name the decimal place of the last digit.	nine thousandths

[latex]-15.571[/latex]
Name the number to the left of the decimal point.	negative fifteen
Write “and” for the decimal point.	negative fifteen and_____
Name the number to the right of the decimal point as if it were a whole number.	negative fifteen and five hundred seventy-one_____
Name the decimal place of the last digit.	negative fifteen and five hundred seventy-one thousandths

six and seventeen hundredths
The word and tells us to place a decimal point.	___.___
The word before and is the whole number; write it to the left of the decimal point.	[latex]6[/latex]._____
The decimal part is seventeen hundredths. Mark two places to the right of the decimal point for hundredths.	[latex]6[/latex]._ _
Write the numerals for seventeen in the places marked.	[latex]6.17[/latex]

fourteen and thirty-seven hundredths
Place a decimal point under the word ‘and’.	______. _________
Translate the words before ‘and’ into the whole number and place it to the left of the decimal point.	[latex]14[/latex]. _________
Mark two places to the right of the decimal point for “hundredths”.	[latex]14[/latex].__ __
Translate the words after “and” and write the number to the right of the decimal point.	[latex]14.37[/latex]
Fourteen and thirty-seven hundredths is written [latex]14.37[/latex].

[latex]4.09[/latex]
There is a [latex]4[/latex] to the left of the decimal point. Write “[latex]4[/latex]” as the whole number part of the mixed number.
Determine the place value of the final digit.
Write the fraction. Write [latex]9[/latex] in the numerator as it is the number to the right of the decimal point.
Write [latex]100[/latex] in the denominator as the place value of the final digit, [latex]9[/latex], is hundredth.	[latex]4{\Large\frac{9}{100}}[/latex]
The fraction is in simplest form.	So, [latex]4.09=4{\Large\frac{9}{100}}[/latex]
Did you notice that the number of zeros in the denominator is the same as the number of decimal places?

[latex]3.7[/latex]
There is a [latex]3[/latex] to the left of the decimal point. Write “[latex]3[/latex]” as the whole number part of the mixed number.
Determine the place value of the final digit.
Write the fraction. Write [latex]7[/latex] in the numerator as it is the number to the right of the decimal point.
Write [latex]10[/latex] in the denominator as the place value of the final digit, [latex]7[/latex], is tenths.	[latex]3{\Large\frac{7}{10}}[/latex]
The fraction is in simplest form.	So, [latex]3.7=3{\Large\frac{7}{10}}[/latex]

[latex]−0.286[/latex]
There is a [latex]0[/latex] to the left of the decimal point. Write a negative sign before the fraction.
Determine the place value of the final digit and write it in the denominator.
Write the fraction. Write [latex]286[/latex] in the numerator as it is the number to the right of the decimal point. Write [latex]1,000[/latex] in the denominator as the place value of the final digit, [latex]6[/latex], is thousandths.	[latex]-{\Large\frac{286}{1000}}[/latex]
We remove a common factor of [latex]2[/latex] to simplify the fraction.	[latex]-{\Large\frac{143}{500}}[/latex]

[latex]18.379[/latex]
Locate the tenths place and mark it with an arrow.
Underline the digit to the right of the tenths digit.
Because [latex]7[/latex] is greater than or equal to [latex]5[/latex], add [latex]1[/latex] to the [latex]3[/latex].
Rewrite the number, deleting all digits to the right of the tenths place.	[latex]18.4[/latex]
So, [latex]18.379[/latex] rounded to the nearest tenth is [latex]18.4[/latex].

[latex]18.379[/latex]
Locate the ones place and mark it with an arrow.
Underline the digit to the right of the ones place.
Since [latex]3[/latex] is not greater than or equal to [latex]5[/latex], do not add [latex]1[/latex] to the [latex]8[/latex].
Rewrite the number, deleting all digits to the right of the ones place.	[latex]18[/latex]
So [latex]18.379[/latex] rounded to the nearest whole number is [latex]18[/latex].

	[latex]0.64[/latex] ____ [latex]0.6[/latex]
Check to see if both numbers have the same number of decimal places. They do not, so write one zero at the right of [latex]0.6[/latex].	[latex]0.64[/latex] ____ [latex]0.6[/latex]
Compare the numbers to the right of the decimal point as if they were whole numbers.	[latex]64>60[/latex]
Order the numbers using the appropriate inequality sign.	[latex]0.64>0.60[/latex] [latex]0.64>0.6[/latex]

	[latex]-0.1[/latex] ____ [latex]- 0.8[/latex]
Write the numbers one under the other, lining up the decimal points.	[latex]-0.1[/latex] [latex]-0.8[/latex]
They have the same number of digits.
Since [latex]-1>-8,-1[/latex] tenth is greater than [latex]-8[/latex] tenths.	[latex]-0.1>-0.8[/latex]

	[latex]x-0.7=1.5[/latex]
Substitute [latex]\color{red}{1}[/latex] for [latex]x[/latex].	[latex]\color{red}{1} - 0.7\stackrel{?}{=}1.5[/latex]
Subtract.	[latex]0.3\not=1.5[/latex]
Since [latex]x=1[/latex] does not result in a true equation, [latex]1[/latex] is not a solution to the equation.

	[latex]x-0.7=1.5[/latex]
Substitute [latex]\color{red}{0.8}[/latex] for [latex]x[/latex].	[latex]\color{red}{0.8} - 0.7\stackrel{?}{=}1.5[/latex]
Subtract.	[latex]-1.5\not=1.5[/latex]
Since [latex]x=-0.8[/latex] does not result in a true equation, [latex]-0.8[/latex] is not a solution to the equation.

		[latex]y+2.3=-4.7[/latex]
Subtract [latex]\color{red}{2.3}[/latex] from each side, to undo the addition.		[latex]y+2.3\color{red}{- 2.3}=-4.7\color{red}{- 2.3}[/latex]
Simplify.		[latex]y-7[/latex]
Check:	[latex]y+2.3=-4.7[/latex]
Substitute [latex]y=\color{red}{-7}[/latex].	[latex]\color{red}{-7}+2.3\stackrel{?}{=}-4.7[/latex]
Simplify.	[latex]-4.7=-4.7[/latex]
Since [latex]y=-7[/latex] makes [latex]y+2.3=-4.7[/latex] a true statement, we know we have found a solution to this equation.

		[latex]a-4.75=-1.39[/latex]
Add [latex]4.75[/latex] to each side, to undo the subtraction.		[latex]a-4.75+\color{red}{4.75}=-1.39+\color{red}{4.75}[/latex]
Simplify.		[latex]a=3.36[/latex]
Check:	[latex]a-4.75=-1.39[/latex]
Substitute [latex]a=\color{red}{3.36}[/latex].	[latex]\color{red}{3.36}-4.75\stackrel{?}{=}-1.39[/latex]
	[latex]-1.39=-1.39[/latex]
Since the result is a true statement, [latex]a=3.36[/latex] is a solution to the equation.

		[latex]-4.8=0.8n[/latex]
We must divide both sides by [latex]0.8[/latex] to isolate [latex]n[/latex].		[latex]{\Large\frac{-4.8}{\color{red}{0.8}}}={\Large\frac{0.8n}{\color{red}{0.8}}}[/latex]
Simplify.		[latex]-6=n[/latex]
Check:	[latex]-4.8=0.8n[/latex]
Substitute [latex]n=\color{red}{-6}[/latex].	[latex]-4.8\stackrel{?}{=}0.8(\color{red}{-6})[/latex]
	[latex]-4.8=-4.8[/latex]
Since [latex]n=-6[/latex] makes [latex]-4.8=0.8n[/latex] a true statement, we know we have a solution.

		[latex]{\Large\frac{p}{-1.8}}=-6.5[/latex]
Here, p is divided by [latex]−1.8[/latex]. We must multiply by [latex]−1.8[/latex] to isolate [latex]p[/latex]		[latex]\color{red}{-1.8}({\Large\frac{p}{-1.8}})=\color{red}{-1.8}(-6.5)[/latex]
Multiply.		[latex]p=11.7[/latex]
Check:	[latex]{\Large\frac{p}{-1.8}}=-6.5[/latex]
	[latex]{\Large\frac{\color{red}{11.7}}{-1.8}}\stackrel{?}{=}-6.5[/latex]
Substitute [latex]p=\color{red}{11.7}[/latex].	[latex]-6.5=-6.5[/latex]
A solution to [latex]{\Large\frac{p}{-1.8}}=-6.5[/latex] is [latex]p=11.7[/latex]

	[latex]x-0.7=1.5[/latex]
Substitute [latex]\color{red}{2.2}[/latex] for [latex]x[/latex].	[latex]\color{red}{2.2} - 0.7\stackrel{?}{=}1.5[/latex]
Subtract.	[latex]1.5=1.5[/latex]
Since [latex]x=2.2[/latex] results in a true equation, [latex]2.2[/latex] is a solution to the equation.

	[latex]0.06x+0.02=0.25x-1.5[/latex]
Multiply both sides by [latex]100[/latex].	[latex]\color{red}{100}(0.06x+0.02)=\color{red}{100}(0.25x-1.5)[/latex]
Distribute.	[latex]100(0.06x)+100(0.02)=100(0.25x)-100(1.5)[/latex]
Multiply, and now no more decimals.	[latex]6x+2=25x-150[/latex]
Collect the variables to the right.	[latex]6x\color{red}{-6x}+2=25x\color{red}{-6x}-150[/latex]
Simplify.	[latex]2=19x-150[/latex]
Collect the constants to the left.	[latex]2\color{red}{+150}=19x-150\color{red}{+150}[/latex]
Simplify.	[latex]152=19x[/latex]
Divide by [latex]19[/latex].	[latex]\Large\frac{152}{\color{red}{19}}\normalsize =\Large\frac{19x}{\color{red}{19}}[/latex]
Simplify.	[latex]8=x[/latex]
Check: Let [latex]x=8[/latex].
[latex]0.06(\color{red}{8})+0.02=0.25(\color{red}{8})-1.5[/latex][latex]0.48+0.02=2.00-1.5[/latex] [latex]0.50=0.50\quad\checkmark[/latex]

	[latex]0.25x+0.05(x+3)=2.85[/latex]
Distribute first.	[latex]0.25x+0.05x+0.15=2.85[/latex]
Combine like terms.	[latex]0.30x+0.15=2.85[/latex]
To clear decimals, multiply by [latex]100[/latex].	[latex]\color{red}{100}(0.30x+0.15)=\color{red}{100}(2.85)[/latex]
Distribute.	[latex]30x+15=285[/latex]
Subtract [latex]15[/latex] from both sides.	[latex]30x+15\color{red}{-15}=285\color{red}{-15}[/latex]
Simplify.	[latex]30x=270[/latex]
Divide by [latex]30[/latex].	[latex]\Large\frac{30x}{\color{red}{30}}\normalsize =\Large\frac{270}{\color{red}{30}}[/latex]
Simplify.	[latex]x=9[/latex]
Check: Let [latex]x=9[/latex].
[latex]0.25x+0.05(x+3)=2.85[/latex][latex]0.25(\color{red}{9})+0.05(\color{red}{9}+3)\stackrel{\text{?}}{=}2.85[/latex] [latex]2.25+0.05(12)\stackrel{\text{?}}{=}2.85[/latex] [latex]2.85=2.85\quad\checkmark[/latex]