We will learn how to simplify the numerical expressions involving fractional numbers. We know how to perform the fundamental operations, namely addition, subtraction, multiplication and division involving fractional numbers and now we will learn to perform two or more operations together.

Solved examples to simplify the numerical expressions involving fractional numbers:

**Simplify the following expression:**

(i) 3\(\frac{3}{4}\) + 3\(\frac{1}{4}\) ÷ 6\(\frac{1}{2}\) - 1\(\frac{1}{4}\)

Solution:

3\(\frac{3}{4}\) + 3\(\frac{1}{4}\) ÷ 6\(\frac{1}{2}\) - 1\(\frac{1}{4}\)

= \(\frac{15}{4}\) + \(\frac{13}{4}\) ÷ \(\frac{13}{2}\) - \(\frac{5}{4}\) (First step: Converting into improper fractions)

= \(\frac{15}{4}\) + \(\frac{13}{4}\) × \(\frac{13}{2}\) - \(\frac{5}{4}\) (Second step: Divide \(\frac{13}{4}\) by \(\frac{13}{2}\))

= \(\frac{15}{4}\) + \(\frac{1}{2}\) - \(\frac{5}{4}\)

= \(\frac{17}{4}\) - \(\frac{5}{4}\)(Third step: Add \(\frac{15}{4}\) + \(\frac{1}{2}\) = \(\frac{17}{4}\))

= \(\frac{12}{4}\)(Fourth step: Subtract \(\frac{17}{4}\) - \(\frac{5}{4}\) = \(\frac{12}{4}\))

= 3 (Fifth step: Reduce the fraction \(\frac{12}{4}\) = 3)

Therefore, 3\(\frac{3}{4}\) + 3\(\frac{1}{4}\) ÷ 6\(\frac{1}{2}\) - 1\(\frac{1}{4}\) = 3

(ii) 3\(\frac{1}{2}\) + 2\(\frac{5}{7}\) × \(\frac{7}{19}\)
- \(\frac{1}{2}\) ÷ 2

**Solution:**

3\(\frac{1}{2}\) + 2\(\frac{5}{7}\) × \(\frac{7}{19}\) - \(\frac{1}{2}\) ÷ 2

= \(\frac{7}{2}\) + \(\frac{19}{7}\) × \(\frac{7}{19}\) - \(\frac{1}{2}\) ÷ 2, (First step: Converting into improper fractions)

= \(\frac{7}{2}\) + \(\frac{19}{7}\) × \(\frac{7}{19}\) - \(\frac{1}{2}\) × \(\frac{1}{2}\), (Second step: Divide \(\frac{1}{2}\) by 2 = \(\frac{1}{2}\) × \(\frac{1}{2}\))

= \(\frac{7}{2}\) + \(\frac{19}{7}\) × \(\frac{7}{19}\) - \(\frac{1}{4}\), (Third step \(\frac{1}{2}\) × \(\frac{1}{2}\) = \(\frac{1}{4}\))

= \(\frac{7}{2}\) + 1 - \(\frac{1}{4}\), (Fourth step: Multiply \(\frac{19}{7}\) × \(\frac{7}{19}\) = 1)

= \(\frac{9}{2}\) - \(\frac{1}{4}\), (Fifth step: Add \(\frac{7}{2}\) + 1 = \(\frac{9}{2}\))

= \(\frac{18 - 1}{4}\), (Sixth step: Subtract \(\frac{9}{2}\) - \(\frac{1}{4}\))

= \(\frac{17}{4}\)

= 4\(\frac{1}{4}\)

Therefore, 3\(\frac{1}{2}\) + 2\(\frac{5}{7}\) × \(\frac{7}{19}\) - \(\frac{1}{2}\) ÷ 2 = 4\(\frac{1}{4}\)

(iii) Simplify: 4\(\frac{1}{7}\) - {2\(\frac{2}{3}\) ÷ (1\(\frac{3}{5}\) - \(\frac{2}{3}\))}

Solution:

4\(\frac{1}{7}\) - {2\(\frac{2}{3}\) ÷ (1\(\frac{3}{5}\) - \(\frac{2}{3}\))}

= \(\frac{29}{7}\) - {\(\frac{8}{3}\) ÷ (\(\frac{8}{5}\) - \(\frac{2}{3}\))} (Converting into proper fractions)

= \(\frac{29}{7}\) - {\(\frac{8}{3}\) ÷ (\(\frac{24 - 10}{15}\))} (Removing round brackets)

= \(\frac{29}{7}\) - {\(\frac{8}{3}\) ÷ \(\frac{14}{15}\)}

= \(\frac{29}{7}\) - {\(\frac{8}{3}\) × \(\frac{15}{14}\)} (Removing curly brackets)

= \(\frac{29}{7}\) - \(\frac{20}{7}\)

= \(\frac{9}{7}\)

= 1\(\frac{2}{7}\)

Therefore, 4\(\frac{1}{7}\) - {2\(\frac{2}{3}\) ÷ (1\(\frac{3}{5}\) - \(\frac{2}{3}\))} = 1\(\frac{2}{7}\).

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