Sure, let's simplify the given expressions step-by-step:
### 2. Simplify the following.
#### Part i: [tex]\(\frac{5}{7} + \frac{-9}{10}\)[/tex]
1. Find a common denominator for the fractions. The denominators are 7 and 10, so their least common multiple (LCM) is 70.
2. Convert each fraction to have the common denominator:
- For [tex]\(\frac{5}{7}\)[/tex]: Multiply the numerator and the denominator by 10, resulting in [tex]\(\frac{5 \times 10}{7 \times 10} = \frac{50}{70}\)[/tex].
- For [tex]\(\frac{-9}{10}\)[/tex]: Multiply the numerator and the denominator by 7, resulting in [tex]\(\frac{-9 \times 7}{10 \times 7} = \frac{-63}{70}\)[/tex].
3. Add the fractions:
[tex]\[
\frac{50}{70} + \frac{-63}{70} = \frac{50 + (-63)}{70} = \frac{-13}{70}
\][/tex]
So, the simplified expression for [tex]\(\frac{5}{7} + \frac{-9}{10}\)[/tex] is [tex]\(\frac{-13}{70}\)[/tex].
To convert this fraction to a decimal, divide the numerator by the denominator:
[tex]\[
\frac{-13}{70} \approx -0.1857
\][/tex]
#### Part iv: [tex]\(-3 + \frac{3}{5}\)[/tex]
1. Convert the whole number to a fraction. [tex]\(-3\)[/tex] can be expressed as [tex]\(\frac{-3}{1}\)[/tex].
2. Find a common denominator for the fractions. The denominators are 1 and 5, so their least common multiple (LCM) is 5.
3. Convert each fraction to have the common denominator:
- For [tex]\(\frac{-3}{1}\)[/tex]: Multiply the numerator and the denominator by 5, resulting in [tex]\(\frac{-3 \times 5}{1 \times 5} = \frac{-15}{5}\)[/tex].
- [tex]\(\frac{3}{5}\)[/tex] already has the denominator 5, so it remains [tex]\(\frac{3}{5}\)[/tex].
4. Add the fractions:
[tex]\[
\frac{-15}{5} + \frac{3}{5} = \frac{-15 + 3}{5} = \frac{-12}{5}
\][/tex]
So, the simplified expression for [tex]\(-3 + \frac{3}{5}\)[/tex] is [tex]\(\frac{-12}{5}\)[/tex].
To convert this fraction to a decimal, divide the numerator by the denominator:
[tex]\[
\frac{-12}{5} = -2.4
\][/tex]
### 3. Verify the following.
The simplified results from above steps are:
1. [tex]\(\frac{5}{7} + \frac{-9}{10} = \frac{-13}{70} \approx -0.1857\)[/tex]
2. [tex]\(-3 + \frac{3}{5} = \frac{-12}{5} = -2.4\)[/tex]
These solutions satisfy the given conditions and calculations.
