Book C: Fractions Assignment - Solutions and Commentary

Introduction

Most modern calculators are able to work with fractions. Fractions can be quite handy for entering in quantities, as well as preserving precision. It is generally easier to enter in a fraction like \( \frac{1}{8} \) than \( 0.125 \). For "recurring" decimals, it is even better. Compare \( \frac{1}{3} \) to \( 0.333333333 \), or \( \frac{1}{7} \) to \( 0.142857142 \).

Here are some useful behavioural rules to go by when using fractions:

• A fraction has the general form \( \frac{d}{c} \), where \( d \) is the numerator, and \( c \) is the denominator. The denominator can be any value except zero.
• Whole numbers can be thought of as fractions with a denominator of 1 i.e. \( a = \frac{a}{1} \).
  
• Calculators will work with proper fractions, where for a fraction \( \frac{b}{c} \), \( b < c \). A fraction larger than one will be represented as \( a \frac{b}{c} \), where \( a \) is an integer, and \( b < c \).
• Calculators will allow you to convert a proper fraction \( a\frac{b}{c} \) into an improper fraction \( \frac{d}{c} \), where \( d > c \).
• Calculators always try to "reduce" fractions to the smallest denominator value possible. For example \( \frac{2}{6} \) will be reduced to \( \frac{1}{3} \). A calculation like \( \frac{1}{3} + \frac{1}{6} \) will give an answer of \( \frac{1}{2} \), not \( \frac{3}{6} \).
• Fractions always have integer numerators and denominators.
• The reciprocal of a fraction \( \frac{d}{c} \) is \( \frac{c}{d} \).
• Multiplying fractions: \( \frac{a}{b} \times \frac{c}{d} = \frac{ab}{cd} \).
• Dividing fractions: \( \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc} \).
• Adding fractions: \( \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd} \).
• Subtracting fractions: \( \frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd} \).
• Calculators will be able to convert back and forth between fractions and numerical expressions. Sometimes this is done by pressing the fraction entry key.
  

Assignment Answers and Commentary

The answers are generally straight conversions between fractions and numerical forms and vice versa.

Items 1 through 4 are the conversion of numerical expressions to fractions. Items 5 through 10 are the conversion of fractions to numerical expressions. All answers are given to 4 sf (significant figures) unless otherwise stated.