Question

adding and subtracting fractions with a different denominator
simplify these expressions without the aid of a calculation, remembering to write the answer in its simplest form.
② $\frac{1}{2} + \frac{4}{5}$
④ $\frac{13}{8} - \frac{3}{5}$
② $\frac{1}{2} + \frac{3}{8} - \frac{1}{4}$
④ $\frac{3}{5} + \frac{3}{10} - \frac{3}{4}$
② $\frac{2}{3} - \frac{1}{4} + \frac{5}{6}$
④ $\frac{7}{12} - \frac{1}{3} + \frac{11}{24}$
adding and subtracting fractions with a different denominator
the same rules apply for questions with a mix of whole numbers and fractions. here are some examples:
simplify these expressions which have a mix of whole numbers and fractions
(i) $3 + \frac{1}{4}$
$3 + \frac{1}{4} = 3\frac{1}{4}$
write the fraction after the whole number
(ii) $1 - \frac{2}{5}$
$1 - \frac{2}{5} = \frac{5}{5} - \frac{2}{5}$
write the whole number as a fraction with same denominator
$= \frac{3}{5}$
subtract the numerators only
(iii) $4 - \frac{2}{7}$
$4 - \frac{2}{7} = \frac{28}{7} - \frac{2}{7}$
write the whole number as a fraction with same denominator
$= \frac{26}{7}$
simplify the fraction
$= 3\frac{5}{7}$
simplify these expressions:
⑧ $2 + \frac{1}{2}$
⑩ $1 + \frac{3}{4}$
② $1 - \frac{2}{3}$
④ $1 - \frac{3}{8}$
② $2 - \frac{3}{5}$
④ $4 - \frac{1}{4}$
⑥ $3 - \frac{5}{3}$
⑧ $5 - \frac{5}{2}$

Explanation:

Let's take the problem \( 2 + \frac{1}{2} \) (from the "Simplify these expressions" section) as an example to solve.

Step1: Recognize the form (whole number + fraction)

We have a whole number \( 2 \) and a fraction \( \frac{1}{2} \). To add them, we can express the whole number as a fraction with the same denominator as the given fraction, but in this case, since we are adding a whole number and a fraction, we can directly combine them into a mixed number. A mixed number has a whole number part and a fractional part.
The whole number is \( 2 \) and the fractional part is \( \frac{1}{2} \), so we write it as \( 2\frac{1}{2} \).
(Alternatively, if we convert the whole number to a fraction: \( 2=\frac{4}{2} \), then \( \frac{4}{2}+\frac{1}{2}=\frac{4 + 1}{2}=\frac{5}{2}=2\frac{1}{2} \))

Step2: No further simplification needed

The mixed number \( 2\frac{1}{2} \) is in its simplest form as the fraction \( \frac{1}{2} \) has no common factors between the numerator and denominator other than \( 1 \).

Step1: Convert whole number to fraction

We know that \( 1=\frac{3}{3} \) (because any whole number \( n \) can be written as \( \frac{n}{1} \), and to get a denominator of \( 3 \), we multiply numerator and denominator by \( 3 \), so \( 1=\frac{3}{3} \)).

Step2: Subtract the fractions

Now we have \( \frac{3}{3}-\frac{2}{3} \). When subtracting fractions with the same denominator, we subtract the numerators and keep the denominator the same. So \( \frac{3-2}{3}=\frac{1}{3} \).

Step1: Convert whole number to fraction with denominator 3

We know that \( 3=\frac{9}{3} \) (since \( 3\times3 = 9 \), so \( \frac{9}{3}=3 \)).

Step2: Subtract the fractions

Now we subtract: \( \frac{9}{3}-\frac{5}{3}=\frac{9 - 5}{3}=\frac{4}{3} \).

Step3: Convert improper fraction to mixed number (optional, but often preferred)

\( \frac{4}{3}=1\frac{1}{3} \) (because \( 3\times1 = 3 \), and \( 4-3 = 1 \), so the whole number part is \( 1 \) and the fractional part is \( \frac{1}{3} \)).

Answer:

\( 2\frac{1}{2} \)

Let's take another example, \( 1-\frac{2}{3} \):