Worksheets Math Grade 2 Fractions. Grade 2 fraction worksheets. Our grade 2 fraction worksheets introduce students to fractions as both parts of a whole and parts of a set.We cover identifying common fractions, comparing common fractions, and reading / writing fractions. Business Math Study Guide 2 – Fractions FB/2015 Page 4 B. TYPES OF FRACTIONS 1. Common Fractions A common fraction is one in which the numerator is less than the denominator (or a fraction which is less than the number 1). A common fraction can also be called a proper fraction. 2 1, 4 3, 93 88, 15 8 are all common fractions.
Learning Outcomes
- Use the least common denominator to eliminate fractions from a linear equation before solving it
- Solve equations with fractions that require several steps
You may feel overwhelmed when you see fractions in an equation, so we are going to show a method to solve equations with fractions where you use the common denominator to eliminate the fractions from an equation. The result of this operation will be a new equation, equivalent to the first, but with no fractions.
Pay attention to the fact that each term in the equation gets multiplied by the least common denominator. That’s what makes it equal to the original!
EXAMPLE
Solve: [latex]frac{1}{8}x+frac{1}{2}=frac{1}{4}[/latex].
Solution:
| [latex]frac{1}{8}x+frac{1}{2}=frac{1}{4}quad{LCD=8}[/latex] | |
| Multiply both sides of the equation by that LCD, [latex]8[/latex]. This clears the fractions. | [latex]color{red}{8(}frac{1}{8}x+frac{1}{2}color{red}{)}=color{red}{8(}frac{1}{4}color{red}{)}[/latex] |
| Use the Distributive Property. | [latex]8cdotfrac{1}{8}x+8cdotfrac{1}{2}=8cdotfrac{1}{4}[/latex] |
| Simplify — and notice, no more fractions! | [latex]x+4=2[/latex] |
| Solve using the General Strategy for Solving Linear Equations. | [latex]x+4color{red}{-4}=2color{red}{-4}[/latex] |
| Simplify. | [latex]x=-2[/latex] |
| Check: Let [latex]x=-2[/latex] [latex]frac{1}{8}x+frac{1}{2}=frac{1}{4}[/latex] [latex]frac{1}{8}(color{red}{-2})+frac{1}{2}stackrel{text{?}}{=}frac{1}{4}[/latex] [latex]frac{-2}{8}+frac{1}{2}stackrel{text{?}}{=}frac{1}{4}[/latex] [latex]frac{-2}{8}+frac{4}{8}stackrel{text{?}}{=}frac{1}{4}[/latex] [latex]frac{2}{8}stackrel{text{?}}{=}frac{1}{4}[/latex] [latex]frac{1}{4}=frac{1}{4}quadcheckmark[/latex] |
In the last example, the least common denominator was [latex]8[/latex]. Now it’s your turn to find an LCD, and clear the fractions before you solve these linear equations.
Notice that once we cleared the equation of fractions, the equation was like those we solved earlier in this chapter. We changed the problem to one we already knew how to solve!
Solve equations by clearing the Denominators
- Find the least common denominator of all the fractions in the equation.
- Multiply both sides of the equation by that LCD. This clears the fractions.
- Isolate the variable terms on one side, and the constant terms on the other side.
- Simplify both sides.
- Use the multiplication or division property to make the coefficient on the variable equal to [latex]1[/latex].
Here’s an example where you have three variable terms. After you clear fractions with the LCD, you will simplify the three variable terms, then isolate the variable.
Example
Solve: [latex]7=frac{1}{2}x+frac{3}{4}x-frac{2}{3}x[/latex].
Show SolutionSolution:
We want to clear the fractions by multiplying both sides of the equation by the LCD of all the fractions in the equation.
| Find the least common denominator of all the fractions in the equation. | [latex]7=frac{1}{2}x+frac{3}{4}x-frac{2}{3}xquad{LCD=12}[/latex] |
| Multiply both sides of the equation by [latex]12[/latex]. | [latex]color{red}{12}(7)=color{red}{12}cdot(frac{1}{2}x+frac{3}{4}x-frac{2}{3}x)[/latex] |
| Distribute. | [latex]12(7)=12cdotfrac{1}{2}x+12cdotfrac{3}{4}x-12cdotfrac{2}{3}x[/latex] |
| Simplify — and notice, no more fractions! | [latex]84=6x+9x-8x[/latex] |
| Combine like terms. | [latex]84=7x[/latex] |
| Divide by [latex]7[/latex]. | [latex]frac{84}{color{red}{7}}=frac{7x}{color{red}{7}}[/latex] |
| Simplify. | [latex]12=x[/latex] |
| Check: Let [latex]x=12[/latex]. | |
| [latex]7=frac{1}{2}x+frac{3}{4}x-frac{2}{3}x[/latex] [latex]7stackrel{text{?}}{=}frac{1}{2}(color{red}{12})+frac{3}{4}(color{red}{12})-frac{2}{3}(color{red}{12})[/latex] [latex]7stackrel{text{?}}{=}6+9-8[/latex] [latex]7=7quadcheckmark[/latex] |
Now here’s a similar problem for you to try. Clear the fractions, simplify, then solve.
Caution!
Sidenotes 1 0 2 Fraction =
One of the most common mistakes when you clear fractions is forgetting to multiply BOTH sides of the equation by the LCD. If your answer doesn’t check, make sure you have multiplied both sides of the equation by the LCD.
In the next example, we’ll have variables and fractions on both sides of the equation. After you clear the fractions using the LCD, you will see that this equation is similar to ones with variables on both sides that we solved previously. Remember to choose a variable side and a constant side to help you organize your work.
Example
Solve: [latex]x+frac{1}{3}=frac{1}{6}x-frac{1}{2}[/latex].
Show SolutionSolution:
| Find the LCD of all the fractions in the equation. | [latex]x+frac{1}{3}=frac{1}{6}x-frac{1}{2},quad{LCD=6}[/latex] |
| Multiply both sides by the LCD. | [latex]color{red}{6}(x+frac{1}{3})=color{red}{6}(frac{1}{6}x-frac{1}{2})[/latex] |
| Distribute. | [latex]6cdot{x}+6cdotfrac{1}{3}=6cdotfrac{1}{6}x-6cdotfrac{1}{2}[/latex] |
| Simplify — no more fractions! | [latex]6x+2=x-3[/latex] |
| Subtract [latex]x[/latex] from both sides. | [latex]6x-color{red}{x}+2=x-color{red}{x}-3[/latex] |
| Simplify. | [latex]5x+2=-3[/latex] |
| Subtract 2 from both sides. | [latex]5x+2color{red}{-2}=-3color{red}{-2}[/latex] |
| Simplify. | [latex]5x=-5[/latex] |
| Divide by [latex]5[/latex]. | [latex]frac{5x}{color{red}{5}}=frac{-5}{color{red}{5}}[/latex] |
| Simplify. | [latex]x=-1[/latex] |
| Check: Substitute [latex]x=-1[/latex]. | |
| [latex]x+frac{1}{3}=frac{1}{6}x-frac{1}{2}[/latex] [latex](color{red}{-1})+frac{1}{3}stackrel{text{?}}{=}frac{1}{6}(color{red}{-1})-frac{1}{2}[/latex] [latex](-1)+frac{1}{3}stackrel{text{?}}{=}-frac{1}{6}-frac{1}{2}[/latex] [latex]-frac{3}{3}+frac{1}{3}stackrel{text{?}}{=}-frac{1}{6}-frac{3}{6}[/latex] [latex]-frac{2}{3}stackrel{text{?}}{=}-frac{4}{6}[/latex] [latex]-frac{2}{3}=-frac{2}{3}quadcheckmark[/latex] |
Now you can try solving an equation with fractions that has variables on both sides of the equal sign. The answer may be a fraction.
In the following video we show another example of how to solve an equation that contains fractions and variables on both sides of the equal sign.
In the next example, we start with an equation where the variable term is locked up in some parentheses and multiplied by a fraction. You can clear the fraction, or if you use the distributive property it will eliminate the fraction. Can you see why?
EXAMPLE
Solve: [latex]1=frac{1}{2}left(4x+2right)[/latex].
Show SolutionSolution:
| [latex]1=frac{1}{2}(4x+2)[/latex] | |
| Distribute. | [latex]1=frac{1}{2}cdot4x+frac{1}{2}cdot2[/latex] |
| Simplify. Now there are no fractions to clear! | [latex]1=2x+1[/latex] |
| Subtract 1 from both sides. | [latex]1color{red}{-1}=2x+1color{red}{-1}[/latex] |
| Simplify. | [latex]0=2x[/latex] |
| Divide by [latex]2[/latex]. | [latex]frac{0}{color{red}{2}}=frac{2x}{color{red}{2}}[/latex] |
| Simplify. | [latex]0=x[/latex] |
| Check: Let [latex]x=0[/latex]. | |
| [latex]1=frac{1}{2}(4x+2)[/latex] [latex]1stackrel{text{?}}{=}frac{1}{2}(4(color{red}{0})+2)[/latex] [latex]1stackrel{text{?}}{=}frac{1}{2}(2)[/latex] [latex]1stackrel{text{?}}{=}frac{2}{2}[/latex] [latex]1=1quadcheckmark[/latex] |
Now you can try solving an equation that has the variable term in parentheses that are multiplied by a fraction.
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| Purpose: | Implements a class for working with rational numbers. |
|---|---|
| Available In: | 2.6 and later |
The Fraction class implements numerical operations for rational numbers based on the API defined by Rational in numbers.
Creating Fraction Instances¶
As with decimal, new values can be created in several ways. One easy way is to create them from separate numerator and denominator values:
The lowest common denominator is maintained as new values are computed.
Another way to create a Fraction is using a string representation of <numerator>/<denominator>:
Strings can also use the more usual decimal or floating point notation of [<digits>].[<digits>].
There are class methods for creating Fraction instances directly from other representations of rational values such as float or decimal.
Notice that for floating point values that cannot be expressed exactly the rational representation may yield unexpected results.
Using decimal representations of the values gives the expected results.
Arithmetic¶
Once the fractions are instantiated, they can be used in mathematical expressions as you would expect.
Approximating Values¶
A useful feature of Fraction is the ability to convert a floating point number to an approximate rational value by limiting the size of the denominator.
See also
- fractions
- The standard library documentation for this module.
- decimal
- The decimal module provides an API for fixed and floating point math.
- numbers
- Numeric abstract base classes.
Sidenotes 1 0 2 Fraction Simplest
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