Question - Fraction Arithmetic
Solution:
The question in the image asks to calculate each expression and simplify the result if possible.Let's solve each expression one by one.a. $$ \frac{3}{3} - \frac{2}{3} \times 7 $$To solve this, we need to follow the order of operations. Multiplication comes before subtraction, so let's first multiply $$ \frac{2}{3} $$ by 7.$$ \frac{2}{3} \times 7 = \frac{2}{3} \times \frac{7}{1} $$$$ = \frac{2 \times 7}{3 \times 1} $$$$ = \frac{14}{3} $$Now subtract this from $$ \frac{3}{3} $$:$$ \frac{3}{3} - \frac{14}{3} = 1 - \frac{14}{3} $$Convert 1 to a fraction with the same denominator to subtract:$$ = \frac{3}{3} - \frac{14}{3} $$$$ = \frac{-11}{3} $$ (which is already in simplest form)b. $$ \frac{2}{5} + \frac{3}{5} \times 2 $$First, we do the multiplication:$$ \frac{3}{5} \times 2 = \frac{3}{5} \times \frac{2}{1} $$$$ = \frac{3 \times 2}{5} $$$$ = \frac{6}{5} $$ or $$ 1\frac{1}{5} $$Now add this to $$ \frac{2}{5} $$:$$ \frac{2}{5} + \frac{6}{5} = \frac{2+6}{5} $$$$ = \frac{8}{5} $$ or $$ 1\frac{3}{5} $$c. $$ \frac{3}{4} + \frac{1}{2} - \frac{2}{5} \times 2 $$Again, we'll start with the multiplication:$$ \frac{2}{5} \times 2 = \frac{2}{5} \times \frac{2}{1} $$$$ = \frac{4}{5} $$Now, before we can add or subtract fractions, we need to have a common denominator. For $$ \frac{3}{4} $$ and $$ \frac{1}{2} $$, the common denominator is 4. For $$ \frac{4}{5} $$, it's not compatible with 4, so we'll keep it separate for now.First, convert $$ \frac{1}{2} $$ to have a denominator of 4:$$ \frac{1}{2} = \frac{1 \times 2}{2 \times 2} $$$$ = \frac{2}{4} $$Now we can add $$ \frac{3}{4} $$ and $$ \frac{2}{4} $$:$$ \frac{3}{4} + \frac{2}{4} = \frac{3+2}{4} = \frac{5}{4} $$ or $$ 1\frac{1}{4} $$We now subtract $$ \frac{4}{5} $$ from $$ \frac{5}{4} $$, but we can't directly subtract since the denominators are different. Instead, we can convert both fractions to have a common denominator, which would be 20 in this case.$$ \frac{5}{4} = \frac{5 \times 5}{4 \times 5} = \frac{25}{20} $$$$ \frac{4}{5} = \frac{4 \times 4}{5 \times 4} = \frac{16}{20} $$Now, subtract the second fraction from the first:$$ \frac{25}{20} - \frac{16}{20} = \frac{25 - 16}{20} $$$$ = \frac{9}{20} $$ (which is already in simplest form)
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