QUESTION TWO Express each of the following as a single, simplified fraction. 2.1. $$ \frac{4}{9}-\frac{2}{3}+\frac{7}{12} $$ 2.2. $$ \frac{-8}{13} imes \frac{5}{24} \div-\frac{25}{13} $$ 2.3. $$ \left(\frac{3}{7}+\frac{1}{4} ight) imes\left(2-\frac{69}{38} ight) $$

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QUESTION TWO Express each of the following as a single, simplified fraction. 2.1. $$ \frac{4}{9}-\frac{2}{3}+\frac{7}{12} $$ 2.2. $$ \frac{-8}{13} 	imes \frac{5}{24} \div-\frac{25}{13} $$ 2.3. $$ \left(\frac{3}{7}+\frac{1}{4}ight) 	imes\left(2-\frac{69}{38}ight) $$

UpStudy AI · Accepted Answer

Let's simplify each of the expressions step by step.

### 2.1. $$ \frac{4}{9}-\frac{2}{3}+\frac{7}{12} $$

First, we need a common denominator for the fractions. The denominators are 9, 3, and 12. The least common multiple (LCM) of these numbers is 36.

Now, we convert each fraction:

- $$ \frac{4}{9} = \frac{4 \times 4}{9 \times 4} = \frac{16}{36} $$
- $$ \frac{2}{3} = \frac{2 \times 12}{3 \times 12} = \frac{24}{36} $$
- $$ \frac{7}{12} = \frac{7 \times 3}{12 \times 3} = \frac{21}{36} $$

Now we can substitute these back into the expression:

$$
\frac{16}{36} - \frac{24}{36} + \frac{21}{36} = \frac{16 - 24 + 21}{36} = \frac{13}{36}
$$

So, the simplified fraction for 2.1 is:

$$
\frac{13}{36}
$$

### 2.2. $$ \frac{-8}{13} \times \frac{5}{24} \div -\frac{25}{13} $$

First, we handle the division by multiplying by the reciprocal:

$$
\frac{-8}{13} \times \frac{5}{24} \times -\frac{13}{25}
$$

Now, we can simplify:

1. The negative signs cancel out.
2. The $$ 13 $$ in the numerator of the second fraction cancels with the $$ 13 $$ in the denominator of the first fraction.

This gives us:

$$
\frac{8 \times 5}{24 \times 25} = \frac{40}{600}
$$

Now, we simplify $$ \frac{40}{600} $$:

$$
\frac{40 \div 40}{600 \div 40} = \frac{1}{15}
$$

So, the simplified fraction for 2.2 is:

$$
\frac{1}{15}
$$

### 2.3. $$ \left(\frac{3}{7}+\frac{1}{4}\right) \times\left(2-\frac{69}{38}\right) $$

First, we simplify $$ \frac{3}{7} + \frac{1}{4} $$:

The common denominator for 7 and 4 is 28.

- $$ \frac{3}{7} = \frac{3 \times 4}{7 \times 4} = \frac{12}{28} $$
- $$ \frac{1}{4} = \frac{1 \times 7}{4 \times 7} = \frac{7}{28} $$

Now we can add these:

$$
\frac{12}{28} + \frac{7}{28} = \frac{19}{28}
$$

Next, we simplify $$ 2 - \frac{69}{38} $$:

Convert 2 to a fraction with a denominator of 38:

$$
2 = \frac{76}{38}
$$

Now we can subtract:

$$
\frac{76}{38} - \frac{69}{38} = \frac{76 - 69}{38} = \frac{7}{38}
$$

Now we multiply the two results:

$$
\frac{19}{28} \times \frac{7}{38} = \frac{19 \times 7}{28 \times 38} = \frac{133}{1064}
$$

Now we simplify $$ \frac{133}{1064} $$:

The GCD of 133 and 1064 is 19.

$$
\frac{133 \div 19}{1064 \div 19} = \frac{7}{56}
$$

Finally, we can simplify $$ \frac{7}{56} $$:

$$
\frac{7 \div 7}{56 \div 7} = \frac{1}{8}
$$

So, the simplified fraction for 2.3 is:

$$
\frac{1}{8}
$$

### Final Answers:
- 2.1: $$ \frac{13}{36} $$
- 2.2: $$ \frac{1}{15} $$
- 2.3: $$ \frac{1}{8} $$
2.1: $$ \frac{13}{36} $$ 2.2: $$ \frac{1}{15} $$ 2.3: $$ \frac{1}{8} $$

QUESTION TWO Express each of the following as a single, simplified fraction. 2.1. \( \frac{4}{9}-\frac{2}{3}+\frac{7}{12} \) 2.2. \( \frac{-8}{13} \times \frac{5}{24} \div-\frac{25}{13} \) 2.3. \( \left(\frac{3}{7}+\frac{1}{4}\right) \times\left(2-\frac{69}{38}\right) \)

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