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Lesson 6 of 9

Fractions · Lesson 6 of 9

Adding Fractions

“Combine equal-sized fractional parts and use equivalent fractions when the original units differ.”

Learning Objectives

• Explain fraction addition with shaded strips and number-line lengths. • Add fractions with equal denominators by counting fractional units. • Find a common denominator before adding unlike fractions. • Apply Brahmagupta’s method to two or more fractions. • Simplify answers and interpret sums in practical situations.

Combine the amounts that were taken

Meena eats half of a chikki and her brother eats one quarter of the same chikki. To find their combined amount, imagine the whole chikki divided into quarters. Meena’s half covers two quarter-pieces and her brother takes one more. Together they have eaten three quarters. Changing the half into quarters lets us count pieces of one size.

Meena: 1/2Brother: 1/4Total: 3/4
Half plus a quarter— The shaded half and the extra quarter together occupy three quarters of the same whole.
Example — chikki eaten and left

Problem
Meena eats 1/2 and her brother eats 1/4 of a whole chikki. How much is eaten, and how much remains?

  1. 1.Represent the half as two quarters: 1/2 = 2/4.
  2. 2.Combine the two quarters with one quarter: 2/4 + 1/4 = 3/4.
  3. 3.Three of the four quarter-pieces are eaten, leaving one quarter. The remaining amount is 1/4 of the whole.

This calculation depends on a consistent whole: both shares refer to the original chikki, not to the amount left after Meena eats. A quarter of the remaining half would be a different quantity. Read the wording carefully and identify the whole before drawing or calculating.

Add counts when the fractional unit matches

Adding fractions with the same denominator is like adding lengths measured in the same unit. Two fifths and one fifth are three fifths because all three pieces are fifth-sized. The denominator stays 5: combining pieces does not change their size. Only the total number of pieces increases.

2/51/53/5
Adding fifths— Count the shaded fifths from both addends: two plus one gives three.
Example — two fifths plus one fifth

Problem
Find 2/5 + 1/5.

  1. 1.Both fractions use the unit 1/5.
  2. 2.There are two units in the first amount and one in the second.
  3. 3.Add the counts: 2 + 1 = 3. The sum is 3/5.

The sum can be greater than one. When four sevenths and six sevenths are combined, there are ten seventh-pieces. Seven of them form a complete whole and three remain. You may report ten sevenths as 10/7 or as the mixed number 1 3/7; both describe exactly the same total.

Example — a sum passing one

Problem
Find 4/7 + 6/7 and express the result as a mixed number.

  1. 1.Both addends count pieces of size 1/7.
  2. 2.Add the counts: 4 + 6 = 10, so the sum is 10/7.
  3. 3.Seven sevenths make one whole. The remaining 10 − 7 = 3 pieces give 3/7.
  4. 4.Thus 4/7 + 6/7 = 10/7 = 1 3/7.
0124/710/7First length: 4/7Next length: 6/7
The same addition on a number line— Start at zero, move four sevenths, then move six more sevenths. The final position is 10/7.

The relationship below records this counting rule. The numerator numbers a and c count units, and the shared denominator b identifies their size. It works for a sum below one, equal to one, or greater than one. With three or more addends, count all the numerators while keeping their common unit.

Addition with one fractional unitLaTeX
a and c are the counts being combined; b is their shared positive denominator.
Do not add denominators

1/5 + 1/5 is 2/5, not 2/10. Two fifth-pieces remain fifth-pieces when combined. Adding the denominators would replace them with smaller pieces and change the quantity.

Different pieces need a common unit

One quarter and one third cannot be counted as two identical pieces because a third is larger than a quarter. We first describe both amounts using an equal-sized fractional unit. A whole divides into twelve twelfths; a quarter then contains three twelfths and a third contains four twelfths. Once rewritten, the units match and ordinary counting works again.

1/4 = 3/121/3 = 4/12Sum = 7/12
Quarters and thirds expressed in twelfths— The first three shaded cells represent a quarter; the next four represent a third. Their total is seven twelfths.
Example — a quarter plus a third

Problem
Find 1/4 + 1/3.

  1. 1.Choose denominator 12, since both 4 and 3 divide it exactly.
  2. 2.Multiply numerator and denominator of 1/4 by 3: 1/4 = 3/12.
  3. 3.Multiply numerator and denominator of 1/3 by 4: 1/3 = 4/12.
  4. 4.Add like-sized units: 3/12 + 4/12 = 7/12. The numbers 7 and 12 have no common factor greater than 1.

This procedure is often associated with Brahmagupta’s description of fraction arithmetic in 628 CE. First choose a common denominator, then write equivalent fractions, combine their numerator counts, and simplify if needed. The product of the denominators always supplies a common multiple. Their lowest common multiple is often smaller and keeps the counting simpler.

Definition
Lowest common multiple

The smallest positive whole number that is a multiple of every number being considered. For denominators 6 and 3, it is 6; for 3 and 5, it is 15.

Brahmagupta’s addition method

• Find a common multiple of the denominators. • Rewrite every fraction with that denominator, preserving its value. • Add the numerators and keep the common denominator. • Reduce the result to lowest terms when possible; write a mixed number if it helps interpret the amount.

Example — two thirds and one fifth

Problem
Find 2/3 + 1/5.

  1. 1.A common denominator is 15.
  2. 2.Change 2/3 to 10/15 by multiplying both numbers by 5.
  3. 3.Change 1/5 to 3/15 by multiplying both numbers by 3.
  4. 4.Add: 10/15 + 3/15 = 13/15. Since 13 and 15 have no common factor greater than 1, this is already in lowest terms.
Example — a smaller denominator is enough

Problem
Find 1/6 + 1/3 and simplify.

  1. 1.Because 6 is a multiple of both 6 and 3, use sixths; using eighteenth-pieces is unnecessary.
  2. 2.Leave 1/6 unchanged and rewrite 1/3 = 2/6.
  3. 3.Add the counts: 1/6 + 2/6 = 3/6.
  4. 4.Divide numerator and denominator by their common factor 3: 3/6 = 1/2. The total is half a whole.

Combine several fractions and interpret the result

A sum may contain more than two fractions. Choose a denominator that works for every addend and keep the original amounts beside their new forms. After adding, simplification can make the answer easier to read. In a real situation, include the measurement unit and check whether the total has the size the situation requires.

Example — three different fractional units

Problem
Find 3/4 + 1/3 + 1/5.

  1. 1.Choose 60, a common multiple of 4, 3, and 5.
  2. 2.Rewrite the fractions as 45/60, 20/60, and 12/60.
  3. 3.Add the counts: 45 + 20 + 12 = 77, giving 77/60.
  4. 4.There is no common factor greater than 1. One whole uses 60 pieces, leaving 17, so the result is also 1 17/60.
Example — mixing paint

Problem
Rahim mixes 2/3 litre of yellow paint and 3/4 litre of blue paint. What is the resulting volume?

  1. 1.Both quantities are measured in litres, so add them.
  2. 2.Use twelfths: 2/3 = 8/12 and 3/4 = 9/12.
  3. 3.The sum is 17/12 litres.
  4. 4.Twelve twelfths make one litre, leaving five twelfths. The total is 1 5/12 litres, assuming the mixture’s volume is the sum of the measured volumes.
Example — enough lace for a border

Problem
Geeta has 2/5 metre of lace and Shamim has 3/4 metre. Is the combined length enough for a border of 1 metre?

  1. 1.Use twentieths: 2/5 = 8/20 and 3/4 = 15/20.
  2. 2.Add: 8/20 + 15/20 = 23/20 metres = 1 3/20 metres.
  3. 3.A one-metre border needs 20/20 metres. They have 23/20, which exceeds that length.
  4. 4.The lace is sufficient, with 3/20 metre beyond the required length.

Quiz

Quick check

What is 2/5 + 1/5?

Quick check

Why does the denominator stay 7 in 4/7 + 6/7?

Quick check

Which mixed number equals 4/7 + 6/7?

Quick check

What is 1/4 + 1/3?

Quick check

The smallest common denominator needed for 1/6 + 1/3 is…

Quick check

The total of 2/5 metre and 3/4 metre is…

Practice Problems

Practice Problems
  1. Use shaded strips and a number line to show 4/7 + 6/7. Explain why both models give the same answer.
  2. Add and simplify: 2/7 + 5/7 + 6/7; 3/4 + 1/3; 2/3 + 5/6; 2/3 + 2/7.
  3. Add 2/3 + 4/5 and then 4/5 + 2/3. Explain why reversing the order does not change the combined amount.
  4. Add 3/5 + 5/8; 9/2 + 5/4; 8/3 + 2/7. Write any result greater than one as a mixed number.
  5. Add 3/4 + 1/3 + 1/5; 2/3 + 4/5 + 3/7; 9/2 + 5/4 + 7/6. Show a common denominator for every sum.
  6. Draw the chikki eaten by Meena and her brother. Explain which whole the fractions refer to and identify the part left.
  7. Explain each step in the paint-mixing and lace-border examples, including the measurement units.
  8. A student writes 1/6 + 1/3 = 2/9. Explain the error with unequal-sized pieces and give the correct sum.

Key Takeaways

Key Takeaways

• Addition combines counts of equal-sized fractional units. • With equal denominators, add numerators and keep the denominator. • With different denominators, first make equivalent fractions using a common denominator. • The same procedure works for more than two addends. • Simplify the total and interpret it with its whole and measurement unit.

Previous · Lesson 5

Comparing and Ordering Fractions

Next · Lesson 7

Subtracting Fractions