Pariksha SaathiLesson 2 of 8 · Mathematics, Classes VI-VIII (Paper II)
Fractions, Decimals, Exponents and Powers, Squares and Cubes
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Fractions
An improper fraction can be written as a mixed fraction (17/5 = 3 2/5), and a fraction is in simplest form when the HCF of its numerator and denominator is 1 (36/48 = 3/4). Unlike fractions are compared through a common denominator or by cross-multiplying: 5/8 < 7/11 because 5 × 11 = 55 is less than 8 × 7 = 56. They are added through the LCM of the denominators: 2 3/5 + 1 3/4 = 52/20 + 35/20 = 87/20 = 4 7/20. 'Of' means multiply, so 3/4 of 16 = 12. Division is multiplication by the reciprocal: 3 1/3 ÷ 2/9 = 10/3 × 9/2 = 15, and a 6 m ribbon gives 6 ÷ 3/4 = 8 pieces of 3/4 m. Dividing by a proper fraction gives a larger answer, which surprises pupils who believe division always makes smaller.
Decimals
Each place to the right of the decimal point is one-tenth of the place before it. Fractions convert through an equivalent fraction with denominator 10, 100 or 1,000, or by division: 7/25 = 28/100 = 0.28 and 3/8 = 0.375. Decimals are compared place by place from the left, so 0.5 > 0.45 although 0.45 has more digits. A product has as many decimal places as its factors together (1.2 × 0.15 = 0.180 = 0.18), and division is easiest once the divisor is a whole number: 2.56 ÷ 0.16 = 256 ÷ 16 = 16.
Laws of exponents
For non-zero a and b and integers m and n: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, aᵐ × bᵐ = (ab)ᵐ, aᵐ ÷ bᵐ = (a/b)ᵐ and a⁰ = 1. Class VIII adds negative exponents, a⁻ᵐ = 1/aᵐ, so 10⁻³ = 1/1,000 and (−2)⁻³ = −1/8. A negative base gives a positive power for an even exponent and a negative one for an odd exponent: (−1)¹⁰¹ = −1. Worked forms: (2⁵ × 2³) ÷ 2⁴ = 2⁴ = 16; (3⁻¹ + 4⁻¹)⁻¹ = (7/12)⁻¹ = 12/7; and if 5ᵐ ÷ 5⁻³ = 5⁵, then m + 3 = 5 and m = 2. Note that (2³)² = 2⁶ = 64, while 2 raised to 3² is 2⁹ = 512.
Standard form
A number is in standard form when written as k × 10ⁿ with 1 ≤ k < 10 and n an integer. Large numbers have a positive n: the speed of light, about 300,000,000 m/s, is 3 × 10⁸ m/s. Small numbers have a negative n: 0.000035 = 3.5 × 10⁻⁵. In standard form the exponent decides first, so 3 × 10⁻⁵ is smaller than 2 × 10⁻⁴. Expanded form with powers of 10 shows place value: 47,561 = 4 × 10⁴ + 7 × 10³ + 5 × 10² + 6 × 10¹ + 1 × 10⁰.
Squares and square roots
A perfect square ends in 0, 1, 4, 5, 6 or 9 and in an even number of zeros, so 2,057 and 1,000 are not squares. The sum of the first n odd numbers is n² (1 + 3 + 5 + 7 + 9 = 25); there are 2n non-square numbers between n² and (n + 1)², so 18 between 81 and 100; and for m > 1, 2m, m² − 1 and m² + 1 form a Pythagorean triplet (m = 4 gives 8, 15, 17). Square roots come from pairing prime factors: 2,352 = 2⁴ × 3 × 7² is not a square because 3 is unpaired, and multiplying by 3 gives 7,056 = 84² while dividing by 3 gives 784 = 28². For decimals, √12.25 = 3.5 and √0.0081 = 0.09.
Cubes and cube roots
The cubes of 1 to 10 are 1, 8, 27, 64, 125, 216, 343, 512, 729 and 1,000. The cube of a negative number is negative, so the cube root of −1,331 is −11. Cube roots come from grouping prime factors in threes: 10,648 = 2³ × 11³, so its cube root is 22, while 675 = 3³ × 5² must be multiplied by 5 to give the cube 3,375 = 15³. In a perfect cube the last digit fixes the root's last digit (a cube ending in 8 has a root ending in 2, 7 gives 3, 3 gives 7, 2 gives 8), and the part left after striking off the last three digits fixes the rest: for 17,576 the root ends in 6 and 17 lies between 2³ = 8 and 3³ = 27, so the root is 26. 1,729 = 1³ + 12³ = 9³ + 10³, the Hardy-Ramanujan number, is the smallest number that is a sum of two positive cubes in two different ways.
In the classroom
Pupils who say 0.45 > 0.5, 1/3 + 1/4 = 2/7 or 2³ = 6 have learnt rules without meaning. Fraction strips and number lines, place-value charts for decimals, squares drawn on grid paper for n² and unit cubes stacked into bigger cubes for n³ tie each symbol to a picture before the shortcut is taught.
Key facts
- Cross-multiplication: 5/8 < 7/11 because 5 × 11 = 55 is less than 8 × 7 = 56.
- Mixed fractions: 2 3/5 + 1 3/4 = 87/20 = 4 7/20, and 3 1/3 ÷ 2/9 = 10/3 × 9/2 = 15.
- Decimals: 3/8 = 0.375; 1.2 × 0.15 = 0.18; 2.56 ÷ 0.16 = 16; 0.5 is greater than 0.45.
- Exponents: (2⁵ × 2³) ÷ 2⁴ = 16; (−2)⁻³ = −1/8; (3⁻¹ + 4⁻¹)⁻¹ = 12/7; (−1)¹⁰¹ = −1.
- (2³)² = 2⁶ = 64, but 2 raised to 3² is 2⁹ = 512; a⁰ = 1 for every non-zero a.
- Standard form: 300,000,000 = 3 × 10⁸ and 0.000035 = 3.5 × 10⁻⁵; in k × 10ⁿ, 1 ≤ k < 10.
- 2,352 = 2⁴ × 3 × 7²: multiplying by 3 gives 7,056 = 84², and dividing by 3 gives 784 = 28².
- Squares: between 9² = 81 and 10² = 100 lie 2 × 9 = 18 non-square numbers; m = 4 in 2m, m² − 1, m² + 1 gives the triplet 8, 15, 17.
- Cube roots: ∛10,648 = 22; ∛17,576 = 26; 675 × 5 = 3,375 = 15³; 1,729 = 1³ + 12³ = 9³ + 10³.
Quick revision
To divide by a fraction multiply by its reciprocal; compare unlike fractions by cross-multiplying; decimal places in a product add up; aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ᵐ = 1/aᵐ; standard form k × 10ⁿ with 1 ≤ k < 10; squares end in 0, 1, 4, 5, 6 or 9; sum of first n odd numbers = n²; 2m, m² − 1, m² + 1 is a triplet; square roots by pairing prime factors and cube roots by grouping them in threes; 1,729 = 1³ + 12³ = 9³ + 10³.