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Magazine Chapter 4: Problem 439
In the following exercises, use a model to find the sum. Draw a picture to illustrate your model. $$1 \frac{5}{6}+1 \frac{5}{6}$$
Short Answer
Expert verified
The sum of 1 \( \frac{5}{6} \) and 1 \( \frac{5}{6} \) is 3 \( \frac{2}{3} \).
Step by step solution
01
- Understanding the Problem
Identify the fractions that need to be added. Here, we have two mixed numbers: 1 \( \frac{5}{6} \) and 1 \( \frac{5}{6} \).
02
- Convert Mixed Numbers to Improper Fractions
Convert each mixed number to an improper fraction. For 1 \( \frac{5}{6} \): 1. Multiply the whole number by the denominator: 1 * 6 = 6.2. Add the numerator: 6 + 5 = 11.3. Place this value over the original denominator: \( \frac{11}{6} \). So, both mixed numbers become \( \frac{11}{6} \).
03
- Add the Fractions
Add the improper fractions: \( \frac{11}{6} + \frac{11}{6} \). Since the denominators are the same, add the numerators: 11 + 11 = 22. The denominator remains 6. Therefore, the result is \( \frac{22}{6} \).
04
- Simplify the Fraction
Simplify \( \frac{22}{6} \). Divide the numerator and the denominator by their greatest common divisor, which is 2: \( \frac{22 \, \div \, 2}{6 \, \div \, 2} = \frac{11}{3} \).
05
- Convert back to a Mixed Number
Convert \( \frac{11}{3} \) back to a mixed number:1. Divide 11 by 3 to get the whole number: 11 \div 3 = 3 with a remainder of 2.2. The quotient is 3 and the remainder forms the numerator over the original denominator: 3 \( \frac{2}{3} \).
06
- Drawing a Model
Draw a picture to illustrate the model:1. Draw two whole circles each divided into 6 equal parts (or rectangles divided the same way).2. Shade 5 out of 6 in each circle to represent \( \frac{5}{6} \).3. Represent adding two circles of \( \frac{5}{6} \) each, showing a total of 3 whole parts and \( \frac{2}{3} \) more.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Fractions
Fractions represent parts of a whole. They consist of two parts: a numerator and a denominator. The numerator is the top number and shows how many parts you have. The denominator is the bottom number and shows how many parts make up a whole.
For example, in the fraction \( \frac{3}{4} \), 3 is the numerator and 4 is the denominator. This means you have 3 out of 4 equal parts of a whole.
Fractions can be added, subtracted, multiplied, or divided by ensuring they share a common denominator when necessary.
For example, in the fraction \( \frac{3}{4} \), 3 is the numerator and 4 is the denominator. This means you have 3 out of 4 equal parts of a whole.
Fractions can be added, subtracted, multiplied, or divided by ensuring they share a common denominator when necessary.
Improper Fractions
Improper fractions occur when the numerator is larger than the denominator. This means you have more than one whole. For instance, \( \frac{9}{5} \) is an improper fraction because 9 is greater than 5.
To convert an improper fraction back to a mixed number:
For example, converting \( \frac{11}{3} \) to a mixed number involves:
1. Dividing 11 by 3, which gives 3 with a remainder of 2.
2. Thus, \( \frac{11}{3} \) becomes \( 3 \frac{2}{3} \).
To convert an improper fraction back to a mixed number:
- Divide the numerator by the denominator. The quotient is the whole number part.
- The remainder forms the new numerator.
For example, converting \( \frac{11}{3} \) to a mixed number involves:
1. Dividing 11 by 3, which gives 3 with a remainder of 2.
2. Thus, \( \frac{11}{3} \) becomes \( 3 \frac{2}{3} \).
Understanding Mixed Numbers
Mixed numbers combine a whole number and a fraction, like \( 1 \frac{5}{6} \). They represent a sum of a whole and a part.
To add mixed numbers, it is often easier to first convert them to improper fractions. Then, perform the addition on these fractions.
For example, to add \( 1 \frac{5}{6} + 1 \frac{5}{6} \):
To add mixed numbers, it is often easier to first convert them to improper fractions. Then, perform the addition on these fractions.
For example, to add \( 1 \frac{5}{6} + 1 \frac{5}{6} \):
- Convert to improper fractions: \( \frac{11}{6} + \frac{11}{6} \).
- Add the fractions: \( \frac{22}{6} \).
- Convert back to a mixed number if needed: \( 3 \frac{2}{3} \).
Using Fraction Models
Fraction models help visualize fractions, mixed numbers, and fraction operations. Common models include pie charts, number lines, or rectangular bars.
In our example of adding \( 1 \frac{5}{6} + 1 \frac{5}{6} \), a model can include:
Using such models makes it easier to grasp how parts combine to form wholes and mixed numbers.
In our example of adding \( 1 \frac{5}{6} + 1 \frac{5}{6} \), a model can include:
- Drawing two circles divided into 6 equal parts.
- Shading 5 out of 6 in each circle to show \( \frac{5}{6} \).
- Combining the circles to illustrate the total sum.
Using such models makes it easier to grasp how parts combine to form wholes and mixed numbers.
