(a) Simplify the expression $3\frac{4}{9} \div (5\frac{1}{3} - 2\frac{3}{4}) + 5\frac{9}{10}$ without using mathematical tables or calculators. (b) A number is selected at random from each of the sets $\{2, 3, 4\}$ and $\{1, 3, 5\}$. Find the probability that the sum of the two numbers is greater than 3 and less than 7.

1. Problem Description

(a) Simplify the expression $3\frac{4}{9} \div (5\frac{1}{3} - 2\frac{3}{4}) + 5\frac{9}{10}$ without using mathematical tables or calculators.

(b) A number is selected at random from each of the sets $\{2, 3, 4\}$ and $\{1, 3, 5\}$. Find the probability that the sum of the two numbers is greater than 3 and less than

7.

2. Solution Steps

(a) Simplify $3\frac{4}{9} \div (5\frac{1}{3} - 2\frac{3}{4}) + 5\frac{9}{10}$:

First, convert the mixed numbers to improper fractions:

$3\frac{4}{9} = \frac{3 \times 9 + 4}{9} = \frac{27 + 4}{9} = \frac{31}{9}$

$5\frac{1}{3} = \frac{5 \times 3 + 1}{3} = \frac{15 + 1}{3} = \frac{16}{3}$

$2\frac{3}{4} = \frac{2 \times 4 + 3}{4} = \frac{8 + 3}{4} = \frac{11}{4}$

$5\frac{9}{10} = \frac{5 \times 10 + 9}{10} = \frac{50 + 9}{10} = \frac{59}{10}$

Now, substitute the improper fractions into the expression:

$\frac{31}{9} \div (\frac{16}{3} - \frac{11}{4}) + \frac{59}{10}$

Find a common denominator for $\frac{16}{3}$ and $\frac{11}{4}$, which is

1

2. $\frac{16}{3} = \frac{16 \times 4}{3 \times 4} = \frac{64}{12}$

$\frac{11}{4} = \frac{11 \times 3}{4 \times 3} = \frac{33}{12}$

$\frac{16}{3} - \frac{11}{4} = \frac{64}{12} - \frac{33}{12} = \frac{64 - 33}{12} = \frac{31}{12}$

Substitute the result back into the expression:

$\frac{31}{9} \div \frac{31}{12} + \frac{59}{10}$

To divide fractions, multiply by the reciprocal:

$\frac{31}{9} \div \frac{31}{12} = \frac{31}{9} \times \frac{12}{31} = \frac{31 \times 12}{9 \times 31} = \frac{12}{9} = \frac{4}{3}$

Substitute the result back into the expression:

$\frac{4}{3} + \frac{59}{10}$

Find a common denominator for $\frac{4}{3}$ and $\frac{59}{10}$, which is

3

0. $\frac{4}{3} = \frac{4 \times 10}{3 \times 10} = \frac{40}{30}$

$\frac{59}{10} = \frac{59 \times 3}{10 \times 3} = \frac{177}{30}$

$\frac{4}{3} + \frac{59}{10} = \frac{40}{30} + \frac{177}{30} = \frac{40 + 177}{30} = \frac{217}{30}$

The simplified expression is $\frac{217}{30}$. We can convert this to a mixed number:

$\frac{217}{30} = 7\frac{7}{30}$

(b) Find the probability that the sum of the two numbers selected from $\{2, 3, 4\}$ and $\{1, 3, 5\}$ is greater than 3 and less than

7. The total number of possible pairs is $3 \times 3 = 9$.

The possible pairs are:

(2, 1), (2, 3), (2, 5)

(3, 1), (3, 3), (3, 5)

(4, 1), (4, 3), (4, 5)

Their sums are:

2 + 1 = 3

2 + 3 = 5

2 + 5 = 7

3 + 1 = 4

3 + 3 = 6

3 + 5 = 8

4 + 1 = 5

4 + 3 = 7

4 + 5 = 9

We want the sums to be greater than 3 and less than

7. This means the sums must be 4, 5, or

6. The pairs that satisfy this condition are:

(2, 3) = 5

(3, 1) = 4

(3, 3) = 6

(4, 1) = 5

There are 4 such pairs.

The probability is $\frac{4}{9}$.

3. Final Answer

(a) $\frac{217}{30}$ or $7\frac{7}{30}$

(b) $\frac{4}{9}$