Fractions, Mixed Numbers & Nearby Round Numbers ഭിന്നസംഖ്യകളും അടുത്ത സംഖ്യകളും
Chapter 4 keeps rewriting numbers: 6½ as 6 + ½, 9½ as 10 − ½, 26 as 30 − 4, and 612 − 170 as 612 − 200 + 30. This page collects the small skills behind those rewrites – mixed numbers, products of fractions, round-number neighbours, subtracting brackets, and the signs that appear when a difference is multiplied.
1. A mixed number is a sum – and also a difference
6½ (ആറര) means 6 + ½. But 9½ is also half less than 10: 9½ = 10 − ½. Both views are useful; the chapter picks whichever makes the arithmetic easy.
The slider moves in quarters. Each number is shown two ways: whole number + fraction and next whole number − fraction.
2. Decimals are fractions in disguise
| Decimal | Fraction | As a sum / difference |
|---|---|---|
| 9.5 | 9½ | 9 + ½ = 10 − ½ |
| 3.5 | 3½ | 3 + ½ |
| 0.25 | ¼ | one quarter |
| 0.75 | ¾ | 1 − ¼ |
| 0.125 | ⅛ | half of a quarter |
| 10.25 | 10¼ | 10 + ¼ |
So 9.5 × 3.5 is the same question as 9½ × 3½, and the chapter treats them alike.
3. Products of fractions
Half of a half is a quarter: ½ × ½ = ¼. In the area picture, a unit square cut in half both ways has four equal pieces. In general (1/a) × (1/b) = 1/(ab).
The shaded cell is 1/a wide and 1/b tall. The unit square holds a × b such cells, so the shaded area is 1/(ab).
½ × ½ = ¼ · ¼ × ¼ = 1/16 · ½ × ⅓ = ⅙ · ¼ × ½ = ⅛ · ½ × 14 = 7 · ¼ × 16 = 4 · ½(x + y) means half of the sum.
4. Nearby round numbers
Multiplying by 30 is easy; multiplying by 26 is not. So write 26 = 30 − 4. Likewise 17 = 20 − 3, 99 = 100 − 1, 48 = 50 − 2, 102 = 100 + 2. The rule: choose the nearest ten (or hundred) and note the signed difference.
5. Subtracting a bracket
Taking away a sum means taking away each part: a − (b + c) = a − b − c. Taking away a difference is subtler: a − (b − c) = a − b + c, because b − c is less than b, so after subtracting b you have removed c too much and must put it back.
a − (b − c) = a − b + c
Textbook trick (page 77): 612 − 170 = 612 − (200 − 30) = 612 − 200 + 30 = 442. Subtracting 200 is easy; then give back the 30 you over-subtracted.
6. Signs when a difference is multiplied
When you multiply (x − y) by (u − v), four products appear and two of them carry minus signs. The rules:
- (+) × (−) = (−): x × (−v) = −xv – removing v from each of x rows removes xv in all.
- (−) × (−) = (+): (−y) × (−v) = +yv – "taking away a removal" puts something back.
Start with the big rectangle xu. Remove the strip xv (right) and the strip yu (bottom). The corner yv was inside both strips, so it was removed twice; add it back once: xu − xv − yu + yv. That is exactly where the "+ yv" in (x − y)(u − v) = xu − xv − yu + yv comes from (Section 4 of the chapter).
Paying ₹170 with two ₹100 notes and getting ₹30 back is 612 − 200 + 30 in action. A cloth merchant selling 9½ m thinks "10 m minus half a metre". A shopkeeper pricing 26 items at ₹17 thinks 30 × 20 and then corrects. Rewriting numbers as convenient sums and differences is everyday arithmetic.
7. Check yourself
- 6½ = 6 + ½ and 9½ = 10 − ½; 9.5 = 9½, 0.25 = ¼, 0.125 = ⅛.
- ½ × ½ = ¼, ¼ × ¼ = 1/16, ½ × ⅓ = ⅙; in general 1/a × 1/b = 1/(ab).
- Use the nearest round number: 26 = 30 − 4, 99 = 100 − 1, 102 = 100 + 2.
- a − (b + c) = a − b − c; a − (b − c) = a − b + c (612 − 170 = 612 − 200 + 30).
- (−y)(−v) = +yv: a corner removed twice is added back once.