Fraction Arithmetic
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)
