Fractions and Decimals
Fractions and Decimals
Fractions and decimals run through the whole paper — in percentage, in mensuration, in data handling. Expect 3 to 4 direct questions and many more that depend on this skill. Speed here decides whether you finish the maths section in time.
Types of Fractions
| Type | Meaning | Example |
|---|---|---|
| Proper | Numerator less than denominator | 3/5 |
| Improper | Numerator greater than or equal | 7/5 |
| Mixed | Whole part plus proper fraction | 1 2/5 |
| Like | Same denominator | 2/7 and 5/7 |
| Unlike | Different denominators | 2/3 and 5/7 |
| Equivalent | Same value, different form | 2/3 and 4/6 |
To compare unlike fractions, either make the denominators equal or cross multiply. For 3/7 and 4/9: 3 × 9 = 27 against 7 × 4 = 28, so 3/7 is the smaller.
Operations on Fractions
a/b ÷ c/d = a/b × d/c
Division by a fraction means multiplication by its reciprocal. This is the rule students accept without understanding, and the paper sometimes asks it in reverse: what must 3/4 be multiplied by to give 1? The answer is its reciprocal, 4/3.
For addition and subtraction of unlike fractions, take the LCM of the denominators — which is why the LCM topic later in this chapter is not a separate skill but the same one.
✗ 2/3 + 3/4 = 5/7 | ✓ 2/3 + 3/4 = 8/12 + 9/12 = 17/12
Decimals and Their Conversion
A decimal is just a fraction with a denominator that is a power of ten. Learn the standard equivalents once and reuse them everywhere.
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 1/8 | 0.125 | 12.5% |
| 1/3 | 0.333... | 33.33% |
| 1/6 | 0.1666... | 16.67% |
| 1/16 | 0.0625 | 6.25% |
A fraction in lowest terms gives a terminating decimal only if its denominator has no prime factor other than 2 and 5. So 3/8 terminates but 5/12 does not. This is a favourite reasoning question and it comes straight from the class 8 chapter.
Multiplying and Dividing Decimals
- Multiply as whole numbers, then place the decimal point so the answer has as many decimal places as both factors together: 0.3 × 0.02 = 0.006.
- To divide by a decimal, shift the point in both numbers until the divisor is whole: 4.5 ÷ 0.5 becomes 45 ÷ 5 = 9.
- Multiplying by a number less than one makes a value smaller, which contradicts most students' intuition.
भिन्न से भाग देना = उसके व्युत्क्रम से गुणा करना।
TRE pointer: Because you will teach classes 6 to 8, the paper likes questions that sit exactly where students stumble: whether multiplying always increases a number, and why dividing by a fraction makes it bigger. The terminating-decimal rule — denominator with only 2s and 5s — is the highest-value single fact in this topic and is rarely prepared properly. Given −1/3 negative marking, memorising the fraction-decimal table saves both marks and minutes.
60-Second Recap
- Proper, improper, mixed, like, unlike and equivalent are the six fraction types.
- Cross multiply to compare two fractions quickly.
- Dividing by a fraction means multiplying by its reciprocal.
- Unlike fractions need the LCM of denominators before adding.
- A fraction terminates as a decimal only if its denominator has no prime factor but 2 and 5.
- Multiplying by a number less than 1 makes the value smaller.
Frequently Asked Questions
Which fractions give terminating decimals?
Only those whose denominator, after reducing the fraction to lowest terms, has no prime factor other than 2 and 5. So 3/8 and 7/20 terminate, while 5/12 and 2/7 do not. This rule appears in the class 8 syllabus and is asked as a reasoning question.
Why does dividing by a fraction give a bigger answer?
Because dividing asks how many of the divisor fit into the dividend, and a fraction smaller than one fits many times. Six divided by one half is twelve, since twelve halves make six. Framing it this way is also the clearest way to teach it to a class 7 student.
How do I compare two fractions quickly?
Cross multiply: for a/b and c/d, compare a times d with b times c. The larger product belongs to the larger fraction. This avoids finding a common denominator and takes only a few seconds, which matters when forty maths questions must be finished in the allotted time.
What is the reciprocal of a mixed number?
Convert the mixed number to an improper fraction first, then invert it. One and two-fifths becomes seven-fifths, whose reciprocal is five-sevenths. Inverting the fractional part alone is a common error and one that examiners plant among the options.