Decimals — Conversion, Operations, and Comparison with Fractions

Question

How do we convert between decimals and fractions, perform operations on decimals, and compare them?

Solution — Step by Step

Count the digits after the decimal point. That tells you the denominator:

1 digit after decimal: denominator = 10
2 digits: denominator = 100
3 digits: denominator = 1000

Examples:

$0.7 = \frac{7}{10}$
$0.25 = \frac{25}{100} = \frac{1}{4}$
$2.375 = \frac{2375}{1000} = \frac{19}{8}$

Always simplify the fraction at the end.

Divide the numerator by the denominator:

$\frac{3}{4} = 3 \div 4 = 0.75$
$\frac{1}{3} = 1 \div 3 = 0.333...$ (repeating)

Quick conversions to memorise: $\frac{1}{2} = 0.5$ , $\frac{1}{4} = 0.25$ , $\frac{3}{4} = 0.75$ , $\frac{1}{5} = 0.2$ , $\frac{1}{8} = 0.125$ . These save time in Class 6-7 exams.

To compare, convert everything to the same form (either all decimals or all fractions):

Comparing decimals: Make them the same length by adding trailing zeros, then compare digit by digit from left:

$0.5$ vs $0.35$ : Write as $0.50$ vs $0.35$ . Since $50 > 35$ , we get $0.5 > 0.35$ .

Comparing decimal with fraction: Convert the fraction to decimal first:

$\frac{2}{5}$ vs $0.45$ : $\frac{2}{5} = 0.4$ , and $0.4 < 0.45$

graph TD
    A[Decimal or Fraction?] --> B{Need to convert?}
    B -->|Decimal to Fraction| C[Count decimal places]
    C --> D[Write digits over 10/100/1000]
    D --> E[Simplify the fraction]
    B -->|Fraction to Decimal| F[Divide numerator by denominator]
    B -->|Compare| G[Convert both to same form]
    G --> H[Compare digit by digit or value by value]

Why This Works

Decimals and fractions are just two ways of writing the same number. A decimal like $0.25$ literally means $\frac{25}{100}$ — the decimal point tells us we are dealing with tenths, hundredths, thousandths. Once we see this connection, conversion becomes mechanical.

The place value system extends naturally: just as the tens place is 10 times the ones place, the tenths place is $\frac{1}{10}$ of the ones place.

Alternative Method

For comparing fractions without converting to decimals, use cross-multiplication:

To compare $\frac{a}{b}$ and $\frac{c}{d}$ : compute $a \times d$ and $b \times c$ .

If $ad > bc$ , then $\frac{a}{b} > \frac{c}{d}$

Example: $\frac{3}{7}$ vs $\frac{2}{5}$ : $3 \times 5 = 15$ and $7 \times 2 = 14$ . Since $15 > 14$ , $\frac{3}{7} > \frac{2}{5}$ .

Common Mistake

Students often think $0.5 < 0.35$ because “35 is bigger than 5.” This is wrong. Always equalise the number of decimal places first: $0.50$ vs $0.35$ . Now it is clear that $50 > 35$ , so $0.5 > 0.35$ . The number of digits after the decimal does NOT determine which number is larger.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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