Columbus State Community College Chapter 4 Section 5 Problem Solving: Mixed Numbers and Estimating

Problem Solving: Mixed Numbers and Estimating Identify mixed numbers and graph them on a number line. Rewrite mixed numbers as improper fractions, or the reverse. Estimate the answer and multiply or divide mixed numbers. Estimate the answer and add or subtract mixed numbers. Solve application problems containing mixed numbers.

Illustrating a Mixed Number with a Diagram EXAMPLE 1 Illustrating a Mixed Number with a Diagram 2 1 4 whole parts shaded is equivalent to shaded parts. 9 1 whole 4 shaded parts 1 whole 4 shaded parts 1 4 of a whole 1 shaded part

Illustrating a Mixed Number with a Number Line EXAMPLE 1 Illustrating Mixed Numbers with a Number Line 8 5 – 3 – 1 is equivalent to 3 5 – 1 8 5 – 8 7 6 5 4 3 2 1 1 2 -2 -1

Mixed Numbers NOTE 1 2 4 represents . 4 + – represents – – , which can also be written as – – . 1 2 4 4 +

Mixed Numbers NOTE In algebra we usually work with the improper fraction form of mixed numbers, especially for negative mixed numbers. However, positive mixed numbers are frequently used in daily life, so it’s important to know how to work with them. For example, we usually say inches rather than inches. 1 2 4 9 2

Writing a Mixed Number as an Improper Fraction Step 1 Multiply the denominator of the fraction times the whole number and add the numerator of the fraction to the product. Step 2 Write the result of Step 1 as the numerator and keep the original denominator. 4 • 5 = 20 3 4 5 20 + 3 = 23 23 4 3 5 =

Writing a Mixed Number as an Improper Fraction EXAMPLE 2 Writing a Mixed Number as an Improper Fraction 2 9 8 Write as an improper fraction. 2 9 8 Step 1 8 • 9 = 72 Step 2 = Then 72 + 2 = 74 2 9 8 74 9  ( 8 • 9 ) + 2 Same denominator

Writing an Improper Fraction as a Mixed Number To write an improper fraction as a mixed number, divide the numerator by the denominator. The quotient is the whole number part (of the mixed number), the remainder is the numerator of the fraction part, and the denominator remains the same. 38 5 Divide 38 by 5. 3 5 7  Remainder 7 Original denominator 5 38 35 3

Writing Improper Fractions as Mixed Numbers EXAMPLE 3 Writing Improper Fractions as Mixed Numbers 28 3 (a) Write as a mixed number. Divide 28 by 3. 1 3 9  Remainder 9 Original denominator 3 28 27 1

Writing Improper Fractions as Mixed Numbers EXAMPLE 3 Writing Improper Fractions as Mixed Numbers 42 9 (b) Write as a mixed number. Divide 42 by 9. 6 9 4  Remainder 2 3 4 4 = Original denominator 9 42 36 Write in lowest terms. 6 9 6

Estimating Mixed Numbers To estimate answers, first round each mixed number to the nearest whole number. If the numerator is half of the denominator or more, round up the whole number part. If the numerator is less than half the denominator, leave the whole number as it is.

Rounding Mixed Numbers to the Nearest Whole Number EXAMPLE 4 Rounding Mixed Numbers to the Nearest Whole Number 4 is more than 1 2 3 4 7 9 (a) Round Half of 7 is 1 2 3 4 7 9 rounds up to 10

Rounding Mixed Numbers to the Nearest Whole Number EXAMPLE 4 Rounding Mixed Numbers to the Nearest Whole Number 3 is less than 4 3 8 6 (b) Round Half of 8 is 4 3 8 6 rounds to 6

Multiplying and Dividing Mixed Numbers Step 1 Rewrite each mixed number as an improper fraction. Step 2 Multiply or divide the improper fractions. Step 3 Write the answer in lowest terms and change it to a mixed number or whole number where possible. This step gives you an answer that is in simplest form.

Estimating the Answer and Multiplying Mixed Numbers EXAMPLE 5 Estimating the Answer and Multiplying Mixed Numbers (a) • 3 4 5 2 Estimate the answer by rounding the mixed numbers. 3 4 5 2 rounds to 4 rounds to 3. and 4 • 3 = 12  Estimated answer

Estimating the Answer and Multiplying Mixed Numbers EXAMPLE 5 Estimating the Answer and Multiplying Mixed Numbers (a) • 3 4 5 2 To find the exact answer, first rewrite each mixed number as an improper fraction. and 3 4 = 15 5 2 14 Step 1 3 7 3 4 • 5 2 = 14 15 Step 2 = 21 2 1 2 10 = 2 1

Estimating the Answer and Multiplying Mixed Numbers EXAMPLE 5 Estimating the Answer and Multiplying Mixed Numbers (a) • 3 4 5 2 Estimate Exact 1 2 10 12 The exact answer is reasonable.

Estimating the Answer and Multiplying Mixed Numbers EXAMPLE 5 Estimating the Answer and Multiplying Mixed Numbers (b) • 1 6 4 7 5 Estimate the answer by rounding the mixed numbers. 1 6 4 7 5 rounds to 1 rounds to 6. and 1 • 6 = 6  Estimated answer

Estimating the Answer and Multiplying Mixed Numbers EXAMPLE 5 Estimating the Answer and Multiplying Mixed Numbers (b) • 1 6 4 7 5 To find the exact answer, first rewrite each mixed number as an improper fraction. and 1 6 = 7 4 5 39 Step 1 1 13 1 6 • 4 7 5 = 39 Step 2 = 13 2 1 2 6 = 2 1

Estimating the Answer and Multiplying Mixed Numbers EXAMPLE 5 Estimating the Answer and Multiplying Mixed Numbers (b) • 1 6 4 7 5 Estimate Exact 1 2 6 6 The exact answer is reasonable.

Estimating the Answer and Dividing Mixed Numbers EXAMPLE 6 Estimating the Answer and Dividing Mixed Numbers (a) ÷ 3 5 7 2 Estimate the answer by rounding the mixed numbers. 3 5 7 2 rounds to 8 rounds to 2. and 8 ÷ 2 = 4  Estimated answer

Estimating the Answer and Dividing Mixed Numbers EXAMPLE 6 Estimating the Answer and Dividing Mixed Numbers (a) ÷ 3 5 7 2 To find the exact answer, first rewrite each mixed number as an improper fraction. and 3 5 7 = 38 2 12 Step 1 19 1 3 5 7 ÷ 2 = 12 38 Step 2 = 5 12 • 38 = 19 6 1 6 1 6 3 =

Estimating the Answer and Dividing Mixed Numbers EXAMPLE 6 Estimating the Answer and Dividing Mixed Numbers (a) ÷ 3 5 7 2 Estimate Exact 1 6 3 4 The exact answer is reasonable.

Estimating the Answer and Dividing Mixed Numbers EXAMPLE 6 Estimating the Answer and Dividing Mixed Numbers (b) ÷ 1 3 2 5 First, round the numbers and estimate the answer. 1 3 2 ÷ 5 2 ÷ 5 Write 2 ÷ 5 using a fraction bar. 2 5

Estimating the Answer and Dividing Mixed Numbers EXAMPLE 6 Estimating the Answer and Dividing Mixed Numbers (b) ÷ 1 3 2 5 Now find the exact answer. and 1 3 2 = 7 5 Step 1 1 3 2 ÷ 5 = 7 Step 2 = 1 5 • 7 3 = 7 15

Estimating the Answer and Dividing Mixed Numbers EXAMPLE 6 Estimating the Answer and Dividing Mixed Numbers (b) ÷ 1 3 2 5 Estimate Exact 2 5 7 15 The estimate and the exact answer are close to one-half. Therefore, the exact answer is reasonable.

Estimating the Answer and Adding Mixed Numbers EXAMPLE 7 Estimating the Answer and Adding Mixed Numbers (a) + 2 3 1 6 5 Estimate the answer by rounding the mixed numbers. 2 3 1 6 5 rounds to 4 rounds to 5. and 4 + 5 = 9  Estimated answer

Estimating the Answer and Adding Mixed Numbers EXAMPLE 7 Estimating the Answer and Adding Mixed Numbers (a) + 2 3 1 6 5 Write your answer in simplest form. To find the exact answer, first rewrite each mixed number as an equivalent improper fraction. Add the numerators. Keep the common denominator. Rewrite each improper fraction with the LCD of 6. 2 3 + 1 6 5 = 31 11 = 31 6 + 22 = 53 6 5 6 8 =

Estimating the Answer and Adding Mixed Numbers EXAMPLE 7 Estimating the Answer and Adding Mixed Numbers (a) + 2 3 1 6 5 Estimate Exact 5 6 8 9 The exact answer is reasonable.

Estimating the Answer and Subtracting Mixed Numbers EXAMPLE 7 Estimating the Answer and Subtracting Mixed Numbers 9 (b) – 4 7 2 Estimate the answer by rounding the mixed numbers. 9 4 7 2 rounds to 9 rounds to 3. and 9 – 3 = 6  Estimated answer

Estimating the Answer and Subtracting Mixed Numbers EXAMPLE 7 Estimating the Answer and Subtracting Mixed Numbers 9 (b) – 4 7 2 To find the exact answer, first rewrite each mixed number as an equivalent improper fraction. Write your answer in simplest form. Rewrite each improper fraction with the LCD of 7. Subtract the numerators. Keep the common denominator. 9 – 4 7 2 = 18 1 = 18 7 – 63 = 45 7 3 7 6 =

Estimating the Answer and Subtracting Mixed Numbers EXAMPLE 7 Estimating the Answer and Subtracting Mixed Numbers 9 (b) – 4 7 2 Estimate Exact 3 7 6 6 The exact answer is reasonable.

+ = Using Your Calculator Try these problems using your calculator. + 5 9 21 1 6 38 1) + c A b c A b 21 5 9 = 13 18 59 c A b c A b 38 1 6

x = Using Your Calculator Try these problems using your calculator. • 4 5 56 8 2) x c A b c A b 56 4 5 = 1 2 35 c A b 5 8

÷ = Using Your Calculator Try these problems using your calculator. ÷ 3 4 83 3) 1 2 11 TI 30X IIS ÷ (-) c A b c A b 83 3 4 = 13 46 7 c A b c A b 11 1 2

÷ = Using Your Calculator Try these problems using your calculator. ÷ 3 4 83 3) 1 2 11 TI 30 Xa ÷ c A b c A b + – 83 3 4 = 13 46 7 c A b c A b 11 1 2

Solving Application Problems with Mixed Numbers EXAMPLE 8 Solving Application Problems: Mixed Numbers (a) Mike started his trip with gallons of gas in his car. After his trip, he had gallons remaining. How many gallons of gas did Mike use on his trip? 3 5 20 1 2 To help understand the mathematical operation needed to solve this problem, read it again using rounded numbers. Mike started his trip with 21 gallons of gas in his car. After his trip he had 6 gallons remaining. How many gallons of gas did Mike use on his trip? 21 – 6 = 15 gallons  Estimate

Solving Application Problems with Mixed Numbers EXAMPLE 8 Solving Application Problems: Mixed Numbers (a) Mike started his trip with gallons of gas in his car. After his trip, he had gallons remaining. How many gallons of gas did Mike use on his trip? 3 5 20 1 2 To find the exact answer, use the original mixed numbers. – c A b c A b 20 3 5 = 1 10 15 c A b c A b 5 1 2 gallons

Solving Application Problems with Mixed Numbers EXAMPLE 8 Solving Application Problems: Mixed Numbers (a) Mike started his trip with gallons of gas in his car. After his trip, he had gallons remaining. How many gallons of gas did Mike use on his trip? 3 5 20 1 2 1 10 15 Mike used gallons of gas on his trip. This result is close to the estimate of 15 gallons.

Solving Application Problems with Mixed Numbers EXAMPLE 8 Solving Application Problems: Mixed Numbers (b) Mary’s recipe for chocolate chip cookies calls for cups of flour per batch. If she has cups of flour available, how many batches of cookies can Mary make? 3 4 15 1 2 To help understand the mathematical operation needed to solve this problem, read it again using rounded numbers. Mary’s recipe for chocolate chip cookies calls for 2 cups of flour per batch. If she has 16 cups of flour available, how many batches of cookies can Mary make? 16 ÷ 2 = 8 batches  Estimate

Solving Application Problems with Mixed Numbers EXAMPLE 8 Solving Application Problems: Mixed Numbers (b) Mary’s recipe for chocolate chip cookies calls for cups of flour per batch. If she has cups of flour available, how many batches of cookies can Mary make? 3 4 15 1 2 To find the exact answer, use the original mixed numbers. ÷ c A b c A b 15 3 4 = c A b c A b 2 1 4 7 batches

Solving Application Problems with Mixed Numbers EXAMPLE 8 Solving Application Problems: Mixed Numbers (b) Mary’s recipe for chocolate chip cookies calls for cups of flour per batch. If she has cups of flour available, how many batches of cookies can Mary make? 3 4 15 1 2 Mary can make 7 batches of cookies. This result is close to the estimate of 8 batches of cookies.

Problem Solving: Mixed Numbers and Estimating Chapter 4 Section 5 – Completed Written by John T. Wallace