Add and Subtract Fractions with Common Denominators

Model Fraction Addition

How many quarters are pictured? One quarter plus \(2\) quarters equals \(3\) quarters.

Remember, quarters are really fractions of a dollar. Quarters are another way to say fourths. So the picture of the coins shows that

\[\begin{array}{lllll}\frac{1}{4} & & \frac{2}{4} & & \frac{3}{4} \\ \text{one quarter} & + & \text{two quarters} & = & \text{three quarters}\end{array}\]

Let’s use fraction circles to model the same example, \(\frac{1}{4}+\frac{2}{4}.\)

Start with one \(\frac{1}{4}\) piece.
Add two more \(\frac{1}{4}\)pieces.
The result is \(\frac{3}{4}\).

So again, we see that

\[\frac{1}{4}+\frac{2}{4}=\frac{3}{4}\]

Example

Try it.

Use a model to find the sum \(\frac{3}{8}+\frac{2}{8}.\)

Solution

Start with three \(\frac{1}{8}\) pieces.
Add two \(\frac{1}{8}\)pieces.
How many \(\frac{1}{8}\)pieces are there?

There are five \(\frac{1}{8}\) pieces, or five-eighths. The model shows that \(\frac{3}{8}+\frac{2}{8}=\frac{5}{8}.\)

Add Fractions with a Common Denominator

shows that to add the same-size pieces, meaning that the fractions have the same denominator, we just add the number of pieces.

Example

Try it.

Find the sum: \(\frac{3}{5}+\frac{1}{5}.\)

Solution

	\(\frac{3}{5}+\frac{1}{5}\)
Add the numerators and place the sum over the common denominator.	\(\frac{3+1}{5}\)
Simplify.	\(\frac{4}{5}\)

Example

Try it.

Find the sum: \(\frac{x}{3}+\frac{2}{3}.\)

Solution

	\(\frac{x}{3}+\frac{2}{3}\)
Add the numerators and place the sum over the common denominator.	\(\frac{x+2}{3}\)

Note that we cannot simplify this fraction any more. Since \(x\) and \(2\) are not like terms, we cannot combine them.

Example

Try it.

Find the sum: \(-\frac{9}{d}+\frac{3}{d}.\)

Solution

We will begin by rewriting the first fraction with the negative sign in the numerator.

\(-\frac{a}{b}=\frac{-a}{b}\)

	\(-\frac{9}{d}+\frac{3}{d}\)
Rewrite the first fraction with the negative in the numerator.	\(\frac{-9}{d}+\frac{3}{d}\)
Add the numerators and place the sum over the common denominator.	\(\frac{-9+3}{d}\)
Simplify the numerator.	\(\frac{-6}{d}\)
Rewrite with negative sign in front of the fraction.	\(-\frac{6}{d}\)

Example

Try it.

Find the sum: \(\frac{2n}{11}+\frac{5n}{11}.\)

Solution

	\(\frac{2n}{11}+\frac{5n}{11}\)
Add the numerators and place the sum over the common denominator.	\(\frac{2n+5n}{11}\)
Combine like terms.	\(\frac{7n}{11}\)

Example

Try it.

Find the sum: \(-\frac{3}{12}+(-\frac{5}{12}).\)

Solution

	\(-\frac{3}{12}+(-\frac{5}{12})\)
Add the numerators and place the sum over the common denominator.	\(\frac{-3+(-5)}{12}\)
Add.	\(\frac{-8}{12}\)
Simplify the fraction.	\(-\frac{2}{3}\)

Model Fraction Subtraction

Subtracting two fractions with common denominators is much like adding fractions. Think of a pizza that was cut into \(12\) slices. Suppose five pieces are eaten for dinner. This means that, after dinner, there are seven pieces (or \(\frac{7}{12}\) of the pizza) left in the box. If Leonardo eats \(2\) of these remaining pieces (or \(\frac{2}{12}\) of the pizza), how much is left? There would be \(5\) pieces left (or \(\frac{5}{12}\) of the pizza).

\[\frac{7}{12}-\frac{2}{12}=\frac{5}{12}\]

Let’s use fraction circles to model the same example, \(\frac{7}{12}-\frac{2}{12}.\)

Start with seven \(\frac{1}{12}\) pieces. Take away two \(\frac{1}{12}\) pieces. How many twelfths are left?

Again, we have five twelfths, \(\frac{5}{12}.\)

Example

Try it.

Use fraction circles to find the difference: \(\frac{4}{5}-\frac{1}{5}.\)

Solution

Start with four \(\frac{1}{5}\) pieces. Take away one \(\frac{1}{5}\) piece. Count how many fifths are left. There are three \(\frac{1}{5}\) pieces left.

Subtract Fractions with a Common Denominator

We subtract fractions with a common denominator in much the same way as we add fractions with a common denominator.

Example

Try it.

Find the difference: \(\frac{23}{24}-\frac{14}{24}.\)

Solution

	\(\frac{23}{24}-\frac{14}{24}\)
Subtract the numerators and place the difference over the common denominator.	\(\frac{23-14}{24}\)
Simplify the numerator.	\(\frac{9}{24}\)
Simplify the fraction by removing common factors.	\(\frac{3}{8}\)

Example

Try it.

Find the difference: \(\frac{y}{6}-\frac{1}{6}.\)

Solution

	\(\frac{y}{6}-\frac{1}{6}\)
Subtract the numerators and place the difference over the common denominator.	\(\frac{y-1}{6}\)

The fraction is simplified because we cannot combine the terms in the numerator.

Example

Try it.

Find the difference: \(-\frac{10}{x}-\frac{4}{x}.\)

Solution

Remember, the fraction \(-\frac{10}{x}\) can be written as \(\frac{-10}{x}.\)

	\(-\frac{10}{x}-\frac{4}{x}\)
Subtract the numerators.	\(\frac{-10-4}{x}\)
Simplify.	\(\frac{-14}{x}\)
Rewrite with the negative sign in front of the fraction.	\(-\frac{14}{x}\)

Now lets do an example that involves both addition and subtraction.

Example

Try it.

Simplify: \(\frac{3}{8}+(-\frac{5}{8})-\frac{1}{8}.\)

Solution

	\(\frac{3}{8}+(-\frac{5}{8})-\frac{1}{8}\)
Combine the numerators over the common denominator.	\(\frac{3+(-5)-1}{8}\)
Simplify the numerator, working left to right.	\(\frac{-2-1}{8}\)
Subtract the terms in the numerator.	\(\frac{-3}{8}\)
Rewrite with the negative sign in front of the fraction.	\(-\frac{3}{8}\)

Key Concepts

• Fraction Addition
  • If \(a,b,\) and \(c\) are numbers where \(c\ne 0\), then \(\frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}\).
  • To add fractions, add the numerators and place the sum over the common denominator.
• Fraction Subtraction
  • If \(a,b,\) and \(c\) are numbers where \(c\ne 0\), then \(\frac{a}{c}-\frac{b}{c}=\frac{a-b}{c}\).
  • To subtract fractions, subtract the numerators and place the difference over the common denominator.

Add and Subtract Fractions with Common Denominators

Model Fraction Addition

In the following exercises, use a model to add the fractions. Show a diagram to illustrate your model.

Try it.

\(\frac{2}{5}+\frac{1}{5}\)

Try it.

\(\frac{3}{10}+\frac{4}{10}\)

Solution

\(\frac{7}{10}\)

Try it.

\(\frac{1}{6}+\frac{3}{6}\)

Try it.

\(\frac{3}{8}+\frac{3}{8}\)

Solution

\(\frac{3}{4}\)

Add Fractions with a Common Denominator

In the following exercises, find each sum.

Try it.

\(\frac{4}{9}+\frac{1}{9}\)

Try it.

\(\frac{2}{9}+\frac{5}{9}\)

Solution

\(\frac{7}{9}\)

Try it.

\(\frac{6}{13}+\frac{7}{13}\)

Try it.

\(\frac{9}{15}+\frac{7}{15}\)

Solution

\(\frac{16}{15}\)

Try it.

\(\frac{x}{4}+\frac{3}{4}\)

Try it.

\(\frac{y}{3}+\frac{2}{3}\)

Solution

\(\frac{y+2}{3}\)

Try it.

\(\frac{7}{p}+\frac{9}{p}\)

Try it.

\(\frac{8}{q}+\frac{6}{q}\)

Solution

\(\frac{14}{q}\)

Try it.

\(\frac{8b}{9}+\frac{3b}{9}\)

Try it.

\(\frac{5a}{7}+\frac{4a}{7}\)

Solution

\(\frac{9a}{7}\)

Try it.

\(\frac{-12y}{8}+\frac{3y}{8}\)

Try it.

\(\frac{-11x}{5}+\frac{7x}{5}\)

Solution

\(\frac{-4x}{5}\)

Try it.

\(-\frac{1}{8}+(-\frac{3}{8})\)

Try it.

\(-\frac{1}{8}+(-\frac{5}{8})\)

Solution

\(-\frac{3}{4}\)

Try it.

\(-\frac{3}{16}+(-\frac{7}{16})\)

Try it.

\(-\frac{5}{16}+(-\frac{9}{16})\)

Solution

\(-\frac{7}{8}\)

Try it.

\(-\frac{8}{17}+\frac{15}{17}\)

Try it.

\(-\frac{9}{19}+\frac{17}{19}\)

Solution

\(\frac{8}{19}\)

Try it.

\(\frac{6}{13}+(-\frac{10}{13})+(-\frac{12}{13})\)

Try it.

\(\frac{5}{12}+(-\frac{7}{12})+(-\frac{11}{12})\)

Solution

\(-\frac{13}{12}\)

Model Fraction Subtraction

In the following exercises, use a model to subtract the fractions. Show a diagram to illustrate your model.

Try it.

\(\frac{5}{8}-\frac{2}{8}\)

Try it.

\(\frac{5}{6}-\frac{2}{6}\)

Solution

\(\frac{1}{2}\)

Subtract Fractions with a Common Denominator

In the following exercises, find the difference.

Try it.

\(\frac{4}{5}-\frac{1}{5}\)

Try it.

\(\frac{4}{5}-\frac{3}{5}\)

Solution

\(\frac{1}{5}\)

Try it.

\(\frac{11}{15}-\frac{7}{15}\)

Try it.

\(\frac{9}{13}-\frac{4}{13}\)

Solution

\(\frac{5}{13}\)

Try it.

\(\frac{11}{12}-\frac{5}{12}\)

Try it.

\(\frac{7}{12}-\frac{5}{12}\)

Solution

\(\frac{1}{6}\)

Try it.

\(\frac{4}{21}-\frac{19}{21}\)

Try it.

\(-\frac{8}{9}-\frac{16}{9}\)

Solution

\(-\frac{8}{3}\)

Try it.

\(\frac{y}{17}-\frac{9}{17}\)

Try it.

\(\frac{x}{19}-\frac{8}{19}\)

Solution

\(\frac{x-8}{19}\)

Try it.

\(\frac{5y}{8}-\frac{7}{8}\)

Try it.

\(\frac{11z}{13}-\frac{8}{13}\)

Solution

\(\frac{11z-8}{13}\)

Try it.

\(-\frac{8}{d}-\frac{3}{d}\)

Try it.

\(-\frac{7}{c}-\frac{7}{c}\)

Solution

\(-\frac{14}{c}\)

Try it.

\(-\frac{23}{u}-\frac{15}{u}\)

Try it.

\(-\frac{29}{v}-\frac{26}{v}\)

Solution

\(-\frac{55}{v}\)

Try it.

\(\frac{6c}{7}-\frac{5c}{7}\)

Try it.

\(\frac{12d}{11}-\frac{9d}{11}\)

Solution

\(\frac{3d}{11}\)

Try it.

\(\frac{-4r}{13}-\frac{5r}{13}\)

Try it.

\(\frac{-7s}{3}-\frac{7s}{3}\)

Solution

\(-\frac{14s}{3}\)

Try it.

\(-\frac{3}{5}-(-\frac{4}{5})\)

Try it.

\(-\frac{3}{7}-(-\frac{5}{7})\)

Solution

\(\frac{2}{7}\)

Try it.

\(-\frac{7}{9}-(-\frac{5}{9})\)

Try it.

\(-\frac{8}{11}-(-\frac{5}{11})\)

Solution

\(-\frac{3}{11}\)

Mixed Practice

In the following exercises, perform the indicated operation and write your answers in simplified form.

Try it.

\(-\frac{5}{18}\cdot \frac{9}{10}\)

Try it.

\(-\frac{3}{14}\cdot \frac{7}{12}\)

Solution

\(-\frac{1}{8}\)

Try it.

\(\frac{n}{5}-\frac{4}{5}\)

Try it.

\(\frac{6}{11}-\frac{s}{11}\)

Solution

\(\frac{6-s}{11}\)

Try it.

\(-\frac{7}{24}+\frac{2}{24}\)

Try it.

\(-\frac{5}{18}+\frac{1}{18}\)

Solution

\(-\frac{2}{9}\)

Try it.

\(\frac{8}{15}\div \frac{12}{5}\)

Try it.

\(\frac{7}{12}\div \frac{9}{28}\)

Solution

\(\frac{49}{27}\)

Condensed: the full section is in OpenStax Prealgebra 2e.

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Øvelse (40)

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1. Simplify: \(2x+9+3x-4.\)

   Arbeid gjennom det Tips

   Vis svaret

   \(5x+5\)

2. Draw a model of the fraction \(\frac{3}{4}.\)

   Arbeid gjennom det Tips

3. Simplify: \(\frac{3+2}{6}.\)

   Arbeid gjennom det Tips

   Vis svaret

   \(\frac{5}{6}\)

4. Use a model to find the sum \(\frac{3}{8}+\frac{2}{8}.\)

   Arbeid gjennom det Tips

   Vis svaret

   Start with three \(\frac{1}{8}\) pieces.
   Add two \(\frac{1}{8}\)pieces.
   How many \(\frac{1}{8}\)pieces are there?

   There are five \(\frac{1}{8}\) pieces, or five-eighths. The model shows that \(\frac{3}{8}+\frac{2}{8}=\frac{5}{8}.\)

5. Use a model to find each sum. Show a diagram to illustrate your model.

   \(\frac{1}{8}+\frac{4}{8}\)

   Arbeid gjennom det Tips

   Vis svaret

   \(\frac{5}{8}\)

6. Use a model to find each sum. Show a diagram to illustrate your model.

   \(\frac{1}{6}+\frac{4}{6}\)

   Arbeid gjennom det Tips

   Vis svaret

   \(\frac{5}{6}\)

7. Find the sum: \(\frac{3}{5}+\frac{1}{5}.\)

   Arbeid gjennom det Tips

   Vis svaret

   	\(\frac{3}{5}+\frac{1}{5}\)
   Add the numerators and place the sum over the common denominator.	\(\frac{3+1}{5}\)
   Simplify.	\(\frac{4}{5}\)

8. Find each sum: \(\frac{3}{6}+\frac{2}{6}.\)

   Arbeid gjennom det Tips

   Vis svaret

   \(\frac{5}{6}\)

9. Find each sum: \(\frac{3}{10}+\frac{7}{10}.\)

   Arbeid gjennom det Tips

   Vis svaret

   1

10. Find the sum: \(\frac{x}{3}+\frac{2}{3}.\)

    Arbeid gjennom det Tips

    Vis svaret

    	\(\frac{x}{3}+\frac{2}{3}\)
    Add the numerators and place the sum over the common denominator.	\(\frac{x+2}{3}\)

    Note that we cannot simplify this fraction any more. Since \(x\) and \(2\) are not like terms, we cannot combine them.

11. Find the sum: \(\frac{x}{4}+\frac{3}{4}.\)

    Arbeid gjennom det Tips

    Vis svaret

    \(\frac{x+3}{4}\)

12. Find the sum: \(\frac{y}{8}+\frac{5}{8}.\)

    Arbeid gjennom det Tips

    Vis svaret

    \(\frac{y+5}{8}\)

13. Find the sum: \(-\frac{9}{d}+\frac{3}{d}.\)

    Arbeid gjennom det Tips

    Vis svaret

    We will begin by rewriting the first fraction with the negative sign in the numerator.

    \(-\frac{a}{b}=\frac{-a}{b}\)

    	\(-\frac{9}{d}+\frac{3}{d}\)
    Rewrite the first fraction with the negative in the numerator.	\(\frac{-9}{d}+\frac{3}{d}\)
    Add the numerators and place the sum over the common denominator.	\(\frac{-9+3}{d}\)
    Simplify the numerator.	\(\frac{-6}{d}\)
    Rewrite with negative sign in front of the fraction.	\(-\frac{6}{d}\)

14. Find the sum: \(-\frac{7}{d}+\frac{8}{d}.\)

    Arbeid gjennom det Tips

    Vis svaret

    \(\frac{1}{d}\)

15. Find the sum: \(-\frac{6}{m}+\frac{9}{m}.\)

    Arbeid gjennom det Tips

    Vis svaret

    \(\frac{3}{m}\)

16. Find the sum: \(\frac{2n}{11}+\frac{5n}{11}.\)

    Arbeid gjennom det Tips

    Vis svaret

    	\(\frac{2n}{11}+\frac{5n}{11}\)
    Add the numerators and place the sum over the common denominator.	\(\frac{2n+5n}{11}\)
    Combine like terms.	\(\frac{7n}{11}\)

17. Find the sum: \(\frac{3p}{8}+\frac{6p}{8}.\)

    Arbeid gjennom det Tips

    Vis svaret

    \(\frac{9p}{8}\)

18. Find the sum: \(\frac{2q}{5}+\frac{7q}{5}.\)

    Arbeid gjennom det Tips

    Vis svaret

    \(\frac{9q}{5}\)

19. Find the sum: \(-\frac{3}{12}+(-\frac{5}{12}).\)

    Arbeid gjennom det Tips

    Vis svaret

    	\(-\frac{3}{12}+(-\frac{5}{12})\)
    Add the numerators and place the sum over the common denominator.	\(\frac{-3+(-5)}{12}\)
    Add.	\(\frac{-8}{12}\)
    Simplify the fraction.	\(-\frac{2}{3}\)

20. Find each sum: \(-\frac{4}{15}+(-\frac{6}{15}).\)

    Arbeid gjennom det Tips

    Vis svaret

    \(-\frac{2}{3}\)

21. Find each sum: \(-\frac{5}{21}+(-\frac{9}{21}).\)

    Arbeid gjennom det Tips

    Vis svaret

    \(-\frac{2}{3}\)

22. Use fraction circles to find the difference: \(\frac{4}{5}-\frac{1}{5}.\)

    Arbeid gjennom det Tips

    Vis svaret

    Start with four \(\frac{1}{5}\) pieces. Take away one \(\frac{1}{5}\) piece. Count how many fifths are left. There are three \(\frac{1}{5}\) pieces left.

23. Use a model to find each difference. Show a diagram to illustrate your model.

    \(\frac{7}{8}-\frac{4}{8}\)

    Arbeid gjennom det Tips

    Vis svaret

    \(\frac{3}{8}\), models may differ.

24. Use a model to find each difference. Show a diagram to illustrate your model.

    \(\frac{5}{6}-\frac{4}{6}\)

    Arbeid gjennom det Tips

    Vis svaret

    \(\frac{1}{6}\), models may differ

25. Find the difference: \(\frac{23}{24}-\frac{14}{24}.\)

    Arbeid gjennom det Tips

    Vis svaret

    	\(\frac{23}{24}-\frac{14}{24}\)
    Subtract the numerators and place the difference over the common denominator.	\(\frac{23-14}{24}\)
    Simplify the numerator.	\(\frac{9}{24}\)
    Simplify the fraction by removing common factors.	\(\frac{3}{8}\)

26. Find the difference: \(\frac{19}{28}-\frac{7}{28}.\)

    Arbeid gjennom det Tips

    Vis svaret

    \(\frac{3}{7}\)

27. Find the difference: \(\frac{27}{32}-\frac{11}{32}.\)

    Arbeid gjennom det Tips

    Vis svaret

    \(\frac{1}{2}\)

28. Find the difference: \(\frac{y}{6}-\frac{1}{6}.\)

    Arbeid gjennom det Tips

    Vis svaret

    	\(\frac{y}{6}-\frac{1}{6}\)
    Subtract the numerators and place the difference over the common denominator.	\(\frac{y-1}{6}\)

    The fraction is simplified because we cannot combine the terms in the numerator.

29. Find the difference: \(\frac{x}{7}-\frac{2}{7}.\)

    Arbeid gjennom det Tips

    Vis svaret

    \(\frac{x-2}{7}\)

30. Find the difference: \(\frac{y}{14}-\frac{13}{14}.\)

    Arbeid gjennom det Tips

    Vis svaret

    \(\frac{y-13}{14}\)

31. Find the difference: \(-\frac{10}{x}-\frac{4}{x}.\)

    Arbeid gjennom det Tips

    Vis svaret

    Remember, the fraction \(-\frac{10}{x}\) can be written as \(\frac{-10}{x}.\)

    	\(-\frac{10}{x}-\frac{4}{x}\)
    Subtract the numerators.	\(\frac{-10-4}{x}\)
    Simplify.	\(\frac{-14}{x}\)
    Rewrite with the negative sign in front of the fraction.	\(-\frac{14}{x}\)

32. Find the difference: \(-\frac{9}{x}-\frac{7}{x}.\)

    Arbeid gjennom det Tips

    Vis svaret

    \(-\frac{16}{x}\)

33. Find the difference: \(-\frac{17}{a}-\frac{5}{a}.\)

    Arbeid gjennom det Tips

    Vis svaret

    \(-\frac{22}{a}\)

34. Simplify: \(\frac{3}{8}+(-\frac{5}{8})-\frac{1}{8}.\)

    Arbeid gjennom det Tips

    Vis svaret

    	\(\frac{3}{8}+(-\frac{5}{8})-\frac{1}{8}\)
    Combine the numerators over the common denominator.	\(\frac{3+(-5)-1}{8}\)
    Simplify the numerator, working left to right.	\(\frac{-2-1}{8}\)
    Subtract the terms in the numerator.	\(\frac{-3}{8}\)
    Rewrite with the negative sign in front of the fraction.	\(-\frac{3}{8}\)

35. Simplify: \(\frac{2}{5}+(-\frac{4}{5})-\frac{3}{5}.\)

    Arbeid gjennom det Tips

    Vis svaret

    −1

36. Simplify: \(\frac{5}{9}+(-\frac{4}{9})-\frac{7}{9}.\)

    Arbeid gjennom det Tips

    Vis svaret

    \(-\frac{2}{3}\)

37. \(\frac{2}{5}+\frac{1}{5}\)

    Arbeid gjennom det Tips

38. \(\frac{3}{10}+\frac{4}{10}\)

    Arbeid gjennom det Tips

    Vis svaret

    \(\frac{7}{10}\)

39. \(\frac{1}{6}+\frac{3}{6}\)

    Arbeid gjennom det Tips

40. \(\frac{3}{8}+\frac{3}{8}\)

    Arbeid gjennom det Tips

    Vis svaret

    \(\frac{3}{4}\)

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Symboler som brukes her

Trykk på et symbol for den fulle definisjonen, et bilde og hva hver bokstav i det betyr.

\neqnot equal
The two sides are different.\pmplus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.\leqless/greater than or equal
Inequalities that allow equality; < and > exclude it.\approxapproximately equal
Equal to the precision shown, not exactly.\sqrt{x}square root, n-th root
The non-negative number whose square (n-th power) is x.a \bmod nremainder
What is left after dividing a by n.\%per cent
Per hundred: 15% = 15/100.a : bratio, fraction
a for every b; a divided by b.

Hvordan: Add and Subtract Fractions with Common Denominators

1. Model fraction addition
2. Add fractions with a common denominator
3. Model fraction subtraction
4. Subtract fractions with a common denominator
5. If
6. To add fractions, add the numerators and place the sum over the common denominator.
7. If
8. To subtract fractions, subtract the numerators and place the difference over the common denominator.

Spørsmål folk stiller

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

Deler av denne siden er tilpasset fra OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). Kondensert og re-forklaret her, feil er vår. Siden kilden er lisensiert for ikke-kommersiell bruk, er denne leksen gratis, selger ingenting, og deles under samme lisens.

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