Mathematics: Class 5 — Chapter 2: Fractions

2.1 Introduction to Fractions

Fractions represent parts of a whole. To compare two fractions of two wholes, the wholes from which the fractions are derived must be the same.

Comparing Wholes of Different Sizes

If we have two chocolates of different sizes, $\frac{1}{3}$ of the larger chocolate can be bigger than $\frac{1}{2}$ of the smaller chocolate. Therefore, we can only say $\frac{1}{2}$ of something is greater than $\frac{1}{3}$ of something when the wholes are identical.

2.2 Playing with a Grid

Using grids to visualize fractions of a group or area:

Shading $\frac{1}{8}$ of Grid A (which has 24 squares, so $\frac{1}{8} \times 24 = 3$ squares).
Shading $\frac{1}{6}$ of Grid B (which has 24 squares, so $\frac{1}{6} \times 24 = 4$ squares).
Shading $\frac{1}{12}$ of Grid C (which has 24 squares, so $\frac{1}{12} \times 24 = 2$ squares).
Finding $\frac{1}{3}$ in the grids (which corresponds to $\frac{1}{3} \times 24 = 8$ squares).

2.3 Equivalent Fractions

Fractions that represent the same part of a whole are called equivalent fractions.

Generating Equivalent Fractions

Equivalent fractions can be generated by multiplying or dividing the numerator and the denominator by the same non-zero number:

$$\frac{1}{2} = \frac{2}{4} = \frac{3}{6} = \frac{4}{8} = \frac{5}{10}$$

$$\frac{1}{3} = \frac{2}{6} = \frac{3}{9} = \frac{4}{12} = \frac{5}{15} = \frac{6}{18} = \frac{8}{24} = \frac{12}{36}$$

$$\frac{2}{5} = \frac{4}{10} = \frac{6}{15} = \frac{8}{20} = \frac{20}{50} = \frac{40}{100}$$

2.4 Comparing Fractions

Comparing Fractions with the Same Denominator

When comparing fractions with the same denominator, the fraction with the larger numerator is greater because it contains more parts of the same size.

Example: Sevi ate $\frac{1}{3}$ of a chikki and Shami ate $\frac{2}{3}$. Since $2 > 1$, we have:

$$\frac{2}{3} > \frac{1}{3}$$

Comparing Fractions with the Same Numerator

When comparing fractions with the same numerator, the fraction with the smaller denominator is greater because the whole is divided into fewer, larger parts.

Example: Comparing $\frac{4}{6}$ and $\frac{4}{5}$ of a paratha. Since a $\frac{1}{6}$ piece is smaller than a $\frac{1}{5}$ piece, 4 pieces of $\frac{1}{6}$ are less than 4 pieces of $\frac{1}{5}$:

$$\frac{4}{6} < \frac{4}{5}$$

2.5 Fractions Greater Than 1

When the numerator is greater than the denominator, the fraction represents a quantity greater than one whole. These can be written as mixed numbers.

Visualizing on a Number Line

We can represent fractions greater than 1 on a number line by dividing each unit interval into equal parts.

5 pieces of $\frac{1}{2}$ paratha: $$\frac{5}{2} = 2 + \frac{1}{2} = 2\frac{1}{2}$$
6 pieces of $\frac{1}{2}$ paratha: $$\frac{6}{2} = 3$$
7 pieces of $\frac{1}{2}$ paratha: $$\frac{7}{2} = 3\frac{1}{2}$$
9 pieces of $\frac{1}{4}$ paratha: $$\frac{9}{4} = 2\frac{1}{4}$$

2.6 Comparing Fractions with Benchmarks

Comparing with Reference to 1 (a whole)

We can compare fractions by checking if they are less than, equal to, or greater than 1.

Example: Compare $\frac{7}{8}$ and $\frac{8}{6}$. Since $\frac{7}{8} < 1$ and $\frac{8}{6} > 1$, we can easily conclude:

$$\frac{7}{8} < \frac{8}{6}$$

Comparing with Reference to $\frac{1}{2}$

We can also use $\frac{1}{2}$ as a benchmark to compare fractions.

Example: Compare $\frac{5}{8}$ and $\frac{3}{6}$. Since $\frac{3}{6} = \frac{1}{2}$ and $\frac{5}{8} > \frac{1}{2}$ (as half of 8 is 4), we have:

$$\frac{5}{8} > \frac{3}{6}$$