Pariksha SaathiLesson 4 of 8 · Mathematics, Classes I-V (Paper I)
Fractions and Decimals at Primary Level: Equivalent Fractions, Comparing, Adding and Money
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Where fractions begin
Children use fraction words long before symbols: half a roti, a quarter of an hour, three-quarters of a glass. NCERT builds on this in Class 4 (Halves and Quarters) and Class 5 (Parts and Wholes) by folding paper, cutting shapes and sharing objects. The key idea is equal parts: a circle cut into 4 unequal pieces does not show quarters. In 3/4 the denominator 4 says how many equal parts the whole is divided into and the numerator 3 says how many of them are taken. A fraction can be part of a whole (a region), part of a collection (3/4 of 20 children), a point on the number line, a measure (1/2 m) or the result of a division: 3 rotis shared equally among 4 children give each child 3/4 of a roti.
Unit fractions and fractions of a collection
A unit fraction has numerator 1. The more parts a whole is cut into, the smaller each part, so 1/8 is smaller than 1/5; this surprises children who know that 8 is bigger than 5, and a sharing story (one cake among 8 friends or among 5) settles it. To find a fraction of a collection, divide by the denominator and multiply by the numerator: 2/5 of 30 = 30 ÷ 5 × 2 = 12. Fractions appear in measurement too: 3/4 of an hour is 45 minutes, 1/4 kg is 250 g and 3/4 m is 75 cm. Fractions less than 1 are proper; those equal to or greater than 1 are improper and can be written as mixed numbers: 7/4 = 1 3/4, because 7 quarters make one whole and 3 quarters more.
Equivalent fractions
Folding a paper strip in half, then in half again and again shows that 1/2 = 2/4 = 4/8: the parts get smaller, but more of them are taken. Multiplying or dividing the numerator and the denominator by the same non-zero number gives an equivalent fraction, and dividing both by their HCF gives the simplest (lowest) form: 18/24 = 3/4 (both divided by 6). Two fractions are equivalent when their cross products are equal: 2/3 and 10/15 are equivalent because 2 × 15 = 3 × 10 = 30. Adding the same number to the numerator and the denominator does not give an equivalent fraction: adding 1 to both terms of 1/2 gives 2/3, which is not equal to 1/2.
Comparing and ordering fractions
With the same denominator (like fractions) the greater numerator gives the greater fraction: 5/9 > 4/9. With the same numerator, the smaller denominator gives the greater fraction: 3/5 > 3/7, because fifths are bigger pieces than sevenths. Otherwise make the denominators equal: 2/3 = 8/12 and 3/4 = 9/12, so 3/4 is greater. Half is a handy benchmark: 3/8 is less than 1/2 (which is 4/8), while 4/7 is more than 1/2 because 4 is more than half of 7. Unit fractions grow as their denominators shrink, so in ascending order they run 1/6, 1/3, 1/2.
Adding and subtracting like fractions
When the pieces are the same size, simply count them: 2/7 + 3/7 = 5/7 and 5/8 - 3/8 = 2/8 = 1/4. The denominator names the size of the piece and does not change, so 2/7 + 3/7 = 5/14 is wrong; a fraction strip shows that 5 sevenths are still sevenths. Taking a part from a whole means writing 1 with the same denominator: if Ravi eats 3/8 of a cake and Sita eats 2/8, together they eat 5/8 and 1 - 5/8 = 3/8 is left. With halves, quarters, thirds and sixths, children can turn one fraction into the other by folding: 1/3 + 1/6 = 2/6 + 1/6 = 3/6 = 1/2.
Decimals from measures and money
The Class 5 chapter Tenths and Hundredths starts from measures children already know. 1 cm = 10 mm, so 1 mm = 1/10 cm = 0.1 cm and 4 cm 7 mm = 4.7 cm; 1 paisa is 1/100 of a rupee, so ₹ 7.75 is 7 rupees 75 paise. The decimal point separates the whole number from the parts, and the places to its right are tenths, hundredths and so on: in 46.27 the 2 is worth 2 tenths (0.2) and the 7 is worth 7 hundredths (0.07). Common fractions convert directly: 1/2 = 0.5, 1/4 = 0.25, 3/4 = 0.75 and 1/10 = 0.1. To compare decimals, give them the same number of places: 0.5 = 0.50, which is greater than 0.45 although 0.45 has more digits, and 2.3 = 2.30.
Money calculations
Rupees and paise are added and subtracted by lining up the decimal points: ₹ 45.50 + ₹ 28.75 = ₹ 74.25, and the change from ₹ 100 is ₹ 25.75. A bill multiplies the rate by the quantity: 3 kg of rice at ₹ 42.50 per kg costs ₹ 127.50. Shop games, price lists and bills give children a reason to handle decimals before they meet the word.
Key facts
- Parts must be equal: a shape cut into 4 unequal pieces does not show quarters; in 3/4, 4 is the denominator (equal parts) and 3 the numerator (parts taken).
- Unit fractions: 1/8 < 1/5, because sharing among more people gives smaller shares.
- Fraction of a collection: 2/5 of 30 = 30 ÷ 5 × 2 = 12; 3/4 of an hour = 45 minutes; 1/4 kg = 250 g.
- Simplest form: 18/24 = 3/4 (divide both terms by their HCF 6); 2/3 = 10/15 because 2 × 15 = 3 × 10.
- Comparing: 5/9 > 4/9 (same denominator), 3/5 > 3/7 (same numerator), and 3/4 > 2/3 because 9/12 > 8/12.
- Like fractions: 2/7 + 3/7 = 5/7, not 5/14; 5/8 - 3/8 = 2/8 = 1/4.
- Fraction as division: 3 rotis shared among 4 children give each 3/4 of a roti; 7/4 = 1 3/4.
- Decimals: 4 cm 7 mm = 4.7 cm; in 46.27 the 7 is 7 hundredths; 0.5 > 0.45 because 0.50 > 0.45.
- Money: ₹ 45.50 + ₹ 28.75 = ₹ 74.25; change from ₹ 100 = ₹ 25.75; 3 kg at ₹ 42.50 per kg = ₹ 127.50.
Quick revision
Fraction = equal parts; denominator = equal parts in the whole, numerator = parts taken; 1/8 < 1/5; 2/5 of 30 = 12; equivalent fractions: multiply or divide both terms by the same non-zero number, 18/24 = 3/4; like fractions: compare and add the numerators only (2/7 + 3/7 = 5/7); 1 mm = 0.1 cm, 1 paisa = ₹ 0.01; 0.5 > 0.45; ₹ 45.50 + ₹ 28.75 = ₹ 74.25.