Two Ways to Find a Common Denominator To add or subtract fractions with different denominators, you must rewrite the fractions with a common denominator. Two Ways to Find a Common Denominator Find the LCM (least common multiple) of the denominators. Multiply the denominators. The LCM of two denominators is the lowest common denominator (LCD) of the fractions. Helpful Hint

Additional Example 2A: Adding and Subtracting Fractions with Unlike Denominators Add. Write the answer in simplest form. 5 6 7 8 + 5 6 7 8 5 · 4 6 · 4 7 · 3 8 · 3 + = + The LCM of the denominator is 24. 20 24 21 24 41 24 17 24 Write equivalent fractions. Add = + = = 1 is a reasonable answer. 1 17 24 Estimate 1 + 1 = 2

Additional Example 2B: Adding and Subtracting Fractions with Unlike Denominators Subtract. Write the answer in simplest form. 2 3 3 4 – 2 3 3 4 2 · 4 3 · 3 – = – Multiply the denominators. 3 · 4 4 · 3 = 8 12 – 9 = – 1 12 Write equivalent fractions. Subtract. is a reasonable answer. - 1 12 Estimate 1 – 1 = 0

Add. Write the answer in simplest form. Additional Example 2C: Adding and Subtracting Fractions with Unlike Denominators Add. Write the answer in simplest form. 2 7 1 3 – + 2 7 + 1 3 = – 2 · 3 7 · 3 + 1 · 7 3 · 7 – Multiply the denominators. = + – 7 21 6 1 21 = Write equivalent fractions. Add Estimate - + = 0 1 2 is a reasonable answer. 1 21

Subtract. Write the answer in simplest form. Check It Out: Example 2B Subtract. Write the answer in simplest form. 2 5 1 2 – 2 5 1 2 2 · 2 1 · 5 – = – Multiply the denominators. 5 · 2 2 · 5 = 4 10 – 5 = 1 10 Write equivalent fractions. Subtract. – Estimate – = 0 1 2 is a reasonable answer. - 1 10

Add. Write the answer in simplest form. Check It Out: Example 2C Add. Write the answer in simplest form. 3 5 1 2 – + 3 5 + 1 2 = – 3 · 2 5 · 2 + 1 · 5 2 · 5 – Multiply the denominators. = + – 5 10 6 1 10 Write equivalent fractions. Add = – Estimate - + = 0 1 2 Is a reasonable answer. - 1 10

Additional Example 3: Astronomy Application In one Earth year, Jupiter completes about of its orbit around the Sun, while Mars completes about of its orbit. How much more of its orbit does Mars complete than Jupiter? 1 2 12 1 2 – = 12 1 · 6 2 · 6 – 1 · 1 12 · 1 The LCM of the denominators is 12. = 6 12 – 1 Write equivalent fractions using the common denominator. = 5 12 Subtract. Mars completes more of its orbit than Jupiter does. 5 12

Subtract the fractions and then subtract the integers. 3 4 5 – 2 1 = Check It Out: Example 3A Subtract. Write the answer in simplest form. 4 5 1 5 3 – 2 Subtract the fractions and then subtract the integers. 3 4 5 – 2 1 = 1 + 3 5 = 1 3 5 Add.

Additional Example 2B: Adding Mixed Numbers Add. Write the answer in simplest form. 1 8 5 6 5 + 3 5 1 8 + 3 6 = 5 + 3 20 24 3 Find a common denominator. = 8 + 23 24 Add the fractions and then add the integers. = 8 23 24 Add.

Check It Out: Example 2B Add. Write the answer in simplest form. 1 4 3 8 4 + 4 4 1 + 4 3 8 = 4 + 4 3 8 2 Find a common denominator. = 8 + 5 8 Add the fractions and then add the integers. = 8 5 8 Add.

REGROUPING MIXED NUMBERS Sometimes when you subtract mixed numbers, the fraction portion of the first number is less than the fraction portion of the second number. In these cases, you must regroup before subtracting. REGROUPING MIXED NUMBERS WORDS NUMBERS Regroup. 1 8 7 = 6 + 1 + Rewrite 1 as a fraction with a common denominator. 8 = 6 + + 1 = 6 9 8 Add.

Remember! Any fraction in which the numerator and denominator are the same is equal to 1.

Additional Example 3C: Subtracting Mixed Numbers Subtract. Write the answer in simplest form. 1 8 3 8 17 – 12 16 9 8 – 12 3 17 1 8 – 12 = 3 Regroup 17 1 8 = 16+ + . = 4 + 6 8 Subtract the fractions and then subtract the integers. = 4 6 8 = 4 3 Add and simplify.

Check It Out: Example 3C Subtract. Write the answer in simplest form. 1 4 3 4 14 – 10 13 5 4 – 10 3 14 1 4 – 10 = 3 Regroup 14 1 4 = 13+ + . = 3 + 2 4 Subtract the fractions and then subtract the integers. = 3 2 4 = 3 1 Add and simplify.

Lesson 2-2C Classwork Textbook p. 100 #1-10 Lesson 2-2D Classwork Textbook p. 106 #1-9