Question

Evaluate (a) \(6 \frac{2}{3} \div 4\) (b) \(10 \div 2 \frac{1}{3}\) (c) \(\left(\frac{1}{2}+\frac{1}{3}\right) \div\left(\frac{2}{3}+\frac{1}{5}\right)\) (d) \(\left(6-2 \frac{1}{3}\right) \times\left(4 \frac{1}{2}-1 \frac{3}{4}\right)\) (e) \(\frac{2 \frac{1}{2}+1 \frac{1}{3}}{6 \frac{2}{3}-2 \frac{1}{4}}\)

Short answer

Question: Evaluate the mixed fraction expressions (a) \(6 \frac{2}{3} \div 4\), (b) \(10 \div 2 \frac{1}{3}\), (c) \(\left(\frac{1}{2}+\frac{1}{3}\right)\div\left(\frac{2}{3}+\frac{1}{5}\right)\), (d) \(\left(6-2 \frac{1}{3}\right) \times\left(4 \frac{1}{2}-1 \frac{3}{4}\right)\), and (e) \(\frac{2 \frac{1}{2}+1 \frac{1}{3}}{6 \frac{2}{3}-2 \frac{1}{4}}\). Answer: (a) \(5\frac{2}{3}\), (b) \(4 \frac{2}{7}\), (c) \(\frac{75}{78}\), (d) \(1\frac{5}{6}\), (e) \(\frac{2}{3}\).

Step-by-step solution

Act 1: Convert mixed fraction to improper fraction

Convert \(6 \frac{2}{3}\) to improper fraction: \(\frac{6*3 + 2}{3} = \frac{20}{3}\). Now, we have: \(\frac{20}{3} \div 4\)

Act 2: Rewrite division as multiplication

Rewrite the expression as multiplication by the reciprocal of 4: \(\frac{20}{3} \times \frac{1}{4}\)

Act 3: Multiply the fractions

Multiply the numerators and denominators: \(\frac{20*1}{3*4} = \frac{20}{12}\)

Act 4: Simplify the fraction

Simplify the fraction by dividing numerator and denominator by their greatest common divisor: \(\frac{20\div4}{12\div4} = \frac{5}{3}\) (a) The answer is \(5\frac{2}{3}\) (mixed fraction form). (b) \(10 \div 2 \frac{1}{3}\)

Act 1: Convert mixed fraction to improper fraction

Convert \(2\frac{1}{3}\) to improper fraction: \(\frac{2 * 3 + 1}{3} = \frac{7}{3}\). Now, we have: \(10 \div \frac{7}{3}\)

Act 2: Rewrite division as multiplication

Rewrite the expression as multiplication by the reciprocal of \(\frac{7}{3}\): \(10 \times \frac{3}{7}\)

Act 3: Multiply the fractions

Multiply \(10 = \frac{10}{1}\) by the fraction: \(\frac{10*3}{1*7} = \frac{30}{7}\) (b) The answer is \(4 \frac{2}{7}\) (mixed fraction form). (c) $\left(\frac{1}{2}+\frac{1}{3}\right) \div\left(\frac{2}{3}+\frac{1}{5}\right)$

Act 1: Add fractions

We have two sets of fractions to add: \(\frac{1}{2}+\frac{1}{3}\) and \(\frac{2}{3}+\frac{1}{5}\). Find a common denominator and add the fractions. \(\frac{1}{2}+\frac{1}{3} = \frac{3+2}{6} = \frac{5}{6}\) \(\frac{2}{3}+\frac{1}{5} = \frac{10+3}{15} = \frac{13}{15}\) Now, we have: \(\frac{5}{6} \div \frac{13}{15}\)

Act 2: Rewrite division as multiplication

Rewrite the expression as multiplication by the reciprocal of \(\frac{13}{15}\): \(\frac{5}{6} \times \frac{15}{13}\)

Act 3: Multiply the fractions

Multiply the numerators and denominators: \(\frac{5*15}{6*13} = \frac{75}{78}\) (c) The answer is \(\frac{75}{78}\) (simplified fraction form). (d) $\left(6-2 \frac{1}{3}\right) \times\left(4 \frac{1}{2}-1 \frac{3}{4}\right)$

Act 1: Subtract mixed fractions

We have two sets of mixed fractions to subtract: \(6 - 2\frac{1}{3}\) and \(4\frac{1}{2}- 1\frac{3}{4}\). Convert the fractions to improper fractions and find a common denominator, and then subtract the fractions. \(6 - 2\frac{1}{3} = \frac{18-7}{3} = \frac{11}{3}\) \(4\frac{1}{2}-1\frac{3}{4}=\frac{9-7}{4} = \frac{2}{4}\) Now, we have: \(\frac{11}{3} \times \frac{2}{4}\)

Act 2: Multiply the fractions

Multiply the numerators and denominators: \(\frac{11*2}{3*4} = \frac{22}{12}\)

Act 3: Simplify the fraction

Simplify the fraction by dividing numerator and denominator by their greatest common divisor: \(\frac{22\div2}{12\div2} = \frac{11}{6}\) (d) The answer is \(1\frac{5}{6}\) (mixed fraction form). (e) \(\frac{2 \frac{1}{2}+1 \frac{1}{3}}{6 \frac{2}{3}-2 \frac{1}{4}}\)

Act 1: Add mixed fractions

We have two sets of mixed fractions to add: \(2\frac{1}{2} + 1\frac{1}{3}\) and \(6\frac{2}{3} - 2\frac{1}{4}\). Convert the fractions to improper fractions and find a common denominator, and then add the fractions. \(2\frac{1}{2}+1\frac{1}{3}=\frac{5+4}{6}=\frac{9}{6}\) \(6\frac{2}{3}-2\frac{1}{4}=\frac{44-9}{12}=\frac{35}{12}\) Now, we have: \(\frac{\frac{9}{6}}{\frac{35}{12}}\)

Act 2: Rewrite division as multiplication

Rewrite the expression as multiplication by the reciprocal of \(\frac{35}{12}\): \(\frac{9}{6} \times \frac{12}{35}\)

Act 3: Multiply the fractions

Multiply the numerators and denominators: \(\frac{9*12}{6*35} = \frac{108}{210}\)

Act 4: Simplify the fraction

Simplify the fraction by dividing numerator and denominator by their greatest common divisor: \(\frac{108\div54}{210\div54} = \frac{2}{3}\) (e) The answer is \(\frac{2}{3}\) (simplified fraction form).

Key concepts

Mixed Numbers

Mixed numbers are a combination of a whole number and a fraction. They are commonly used when you have more than one whole and a fractional part together. For example, in the exercise, you see numbers like \(6\frac{2}{3}\), which indicates six whole units and two-thirds of another unit. When performing arithmetic operations with mixed numbers, it’s often easier to convert them to improper fractions first.
This involves multiplying the whole number by the fraction's denominator and adding the numerator. The result becomes the new numerator over the original denominator. So, for \(6\frac{2}{3}\), convert it by doing \(6 \times 3 + 2 = 20\), giving the improper fraction \(\frac{20}{3}\).

Improper Fractions

An improper fraction is a fraction where the numerator is greater than or equal to the denominator, meaning it represents a value equal to or greater than one whole. They are often easier to work with, especially in multiplication and division. For example, once we've converted \(6\frac{2}{3}\) to \(\frac{20}{3}\), it’s easier to handle arithmetic operations.To convert mixed numbers into improper fractions, multiply the whole number by the denominator and add the result to the numerator. The sum is then placed over the original denominator, making calculations simpler and more straightforward.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form where the numerator and denominator share no common divisors other than 1. This step ensures the fraction is as concise as possible. In our example, after multiplying fractions, we arrived at \(\frac{20}{12}\).
To simplify, you find the greatest common divisor (GCD) of both numbers. Here, the GCD is 4, so divide both the numerator and denominator by 4 to get \(\frac{5}{3}\). Simplification helps make fractions easier to understand and compare.

Division of Fractions

Dividing fractions might seem complex at first, but it follows a simple rule: multiply by the reciprocal. The reciprocal of a fraction is obtained by swapping its numerator and denominator. For example, if you have to calculate \(\frac{20}{3} \div 4\), first express 4 as \(\frac{4}{1}\) then use the reciprocal to rewrite the operation: \(\frac{20}{3} \times \frac{1}{4}\).
Performing the multiplication will then give \(\frac{20 \times 1}{3 \times 4} = \frac{20}{12}\), which can be simplified further. By following the reciprocal rule, you can turn complex division into a simple multiplication problem.