WBBSE Solutions For Class 10 Maths Algebra Chapter 2 Ratio And Proportion

Algebra Chapter 2 Ratio And Proportion

⇔ How many times or division of another same quantity in comparison to a quantity is called ratio.

⇒ \(a: b=\frac{a}{b}=\frac{a k}{b k}=a k: b k,(k \neq 0)\)

⇒ or, \(\quad a: b=\frac{\frac{a}{k}}{\frac{b}{k}}=\frac{a}{k}: \frac{b}{k}(k \neq 0)\)

⇒ That means a ratio does not alter if its first and 2nd terms are multiplied or divided by the same nonzero number.

⇒ The value of ratio of two real numbers x and y (y≠ 0) is x: y or \(\frac{x}{y}\), this x: y is read as ‘x is to y’, x is called the antecedent and y is called consequent of the ratio.

Class 10 Maths Algebra Chapter 2 Solutions

⇒ If \(\frac{x}{y}\) > 1 then it is called a ratio of greater inequality and if \(\frac{x}{y}\) < 1, the ratio is called the ratio of less inequality.

⇒ If x = y then the ratio is called the ratio of equality.

⇒ y: x is the inverse ratio of x: y.

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⇒ From two or more given ratios, if we take antecedent as a product of antecedents of the given ratios and consequent as a product of consequents of the given ratios, then the ratio is called mixed or compound ratio.

⇒ The compound ratio of a: b, c : d, and e: f is ace: bdf.

Proportion:

If four real numbers are such that the ratio of the first two is equal to ratio of the last two, then the four numbers in that order are said to be proportional or said to be in proportion.

⇒ If four real numbers p, q, r, s (q ≠ 0, s ≠ 0) be in proportion, we write p: q : : r: s, where p, q are called extreme terms; q, r are called middle terms and s is called the fourth term.

⇒ If a: b : : b: c then a, b, c are in continued proportion, and b is called the mean proportional of a & c.

⇒ Here b² = ac or b = ±√ac

Properties of proportion:

1. If a: b : : c : d then a: c : : b : d.

⇒ This property is called alternendo.

2. If a: b : : c : d then b: a : : d: c

⇒ This property is called invertendo.

3. If a : b : : c : d then (a + b) : b : : (c + d) : d

⇒ This property is called componendo.

4. If (a: b : : c : d then (a – b): b : : (c – d) : d.

⇒ This property is called Dividendo.

5. If a: b :: c : d then \(\frac{a + b}{a – b}\) : : \(\frac{c + d}{c – d}\)

⇒ This property is culled Componendo And Dividendo.

6. If a: b : : c : d then each ratio = \(\frac{a + c}{b + d}\)

⇒ This property is called Addendo.

Quadratic Equations Class 10 Solutions

Algebra Chapter 2 Ratio And Proportion True Or False

Example 1. The mean proportional of 4 and 16 is ±.8.

Solution: True

Example 2. In any ratio of greater inequality antecedent > consequent.

Solution: True

Example 3. In any ratio of less inequality antecedent = consequent.

Solution: False

Example 4. If product of three positive continued proportional number is 27, then their mean proportion is 9.

Solution: False

Example 5. Mixed ratio of \(x: \frac{y z}{x}, y: \frac{z x}{y} \text { and } z: \frac{x y}{z}\) is 1: 1.

Solution: True

Class 10 Algebra Chapter 2 Solved Examples

Example 6. The compound ratio of ab: c², bc: a^2, and ca: b² is 1: 1.

Solution: True

Example 7. x³y, x²y²y, and xy² are in continued proportion.

Solution: True

Example 8. Fourth proportional of 2, 6, and 8 is 16.

Solution: False

Example 9. If x, y, and z are in continued proportion then xy = z².

Solution: False

Example 10. If x : y = 5 : 7 then \(\frac{1}{x^2}: \frac{1}{y^2}\) = 49 : 25

Solution: True

Algebra Chapter 2 Ratio And Proportion Fill In The Blanks

Example 1. If the product of three positive continued proportional number is 64, then their mean proportional is ______

Solution: 4

Example 2. If a : 2 = b : 5 = c : 8 then 50% of a = 20% of b = ________% of c.

Solution: 12\(\frac{1}{2}\)

Example 3. The mean proportional of (x – 2) and (x- 3) is x, then value of x is _______

Solution: \(\frac{6}{5}\)

Example 4. If the value of a ratio is greater than 1, then the ratio is called ______

Solution: Greater inequality

Quadratic Equations Formulas Class 10

Example 5. If the value of a ratio is less than 1, then the ratio is called _______

Solution: Less inequality

Example 6. The inverse ratio of x : 3 is ______

Solution: 3: 2

Example 7. 5: 5 is called the ratio of ______

Solution: Equality

Example 8. If x: 2x : : 3: y then y = ______

Solution: 6

Example 9. If a, b, c, and d are in proportion then ad = ______

Solution: bc

Class 10 Maths Algebra Important Questions

Example 10. Third proportion of Rs 4 and Rs 12 is _______

Solution: Rs. 36

Example 11. If a, b, and c are in continued proportion then b is called the _______ and c is called the ______

Solution: Mean proportion, third proportion

Example 12. If A : B = 5 : 6, B:C = 6:7, C:D = 7:8, D:E = 8:5 then A : E = ______

Solution: 1: 1

Algebra Chapter 2 Ratio And Proportion Short Answer Type Questions

Example 1. If \(\frac{a}{2}=\frac{b}{3}=\frac{c}{4}=\frac{2 a-3 b+4 c}{p}\), find p.

Solution: Let \(\frac{a}{2}=\frac{b}{3}=\frac{c}{4}=k \quad(\neq 0)\)

∴ \(\frac{2 a-3 b+4 c}{p}=\frac{2 \cdot 2 k-3 \cdot 3 k+4 \cdot 4 k}{p}=k\)

⇒ or, \(\frac{11 k}{p}=k\) ∴ p=11

Example 2. If \(\frac{3 x-5 y}{3 x+5 y}=\frac{1}{2}\) find the value of \(\frac{3 x^2-5 y^2}{3 x^2+5 y^2}\)

Solution: \(\frac{3 x-5 y+3 x+5 y}{3 x-5 y-3 x-5 y}=\frac{1+2}{1-2}\)

⇒ or, \(\frac{6 x}{-10 y}=\frac{3}{-1}\) or, \(\frac{3 x}{5 y}=3\) or, x = 5y

⇒ Now, \(\frac{3 x^2}{5 y^2}=\frac{3 \cdot 25 y^2}{5 y^2}=15\)

⇒ ∴ \(\frac{3 x^2-5 y^2}{3 x^2+5 y^2}=\frac{15-1}{15+1}=\frac{14}{16}=\frac{7}{8}\)

Class 10 Maths Algebra Important Questions

Example 3. If a : A = 3 : 4, x : y = 5 : 7 then find the value of (3ax – by) : (4by – 7ax)

Solution: \(\frac{a}{b} \times \frac{x}{y}=\frac{3}{4} \times \frac{5}{7}\)

⇒ or, \(\frac{a x}{b y}=\frac{15}{28}\)

⇒ ∴ \(\frac{3 a x-b y}{4 b y-7 a x}=\frac{3 \times \frac{15 b y}{28}-b y}{4 b y-7 \times \frac{15 b y}{28}}\)

= \(\frac{4\left(\frac{+17 b y}{28}\right)}{b y}=\frac{17}{7}\)

Example 4. If x, 12, y, and 27 are in continued proportion, find the positive value of x & y.

Solution: y² = 12 x 27

⇒ or, y = \(\sqrt{12 \times 27}\) = 18

⇒ Now, xy = 12²

⇒ or, x = \(\frac{144}{18}\) = 8

Example 5. If a: b = 3: 2, b: c = 3: 2, then find the value of (a + b) : (b + c).

Solution: a: b = 3: 2 = 9: 6

⇒ b: c = 3: 2 = 6: 4

⇒ ∴ a : b : c = 9 : 6 : 4

⇒ Let, a, b, c be 9k, 6k, 4k (k ≠ 0)

⇒ \(\frac{a+b}{b+c}=\frac{9 k+6 k}{6 k+4 k}=\frac{15}{10}=\frac{3}{2}\)

Class 10 Maths Algebra Important Questions

Example 6. If a, b, c, d are in continued proportion, then find the value of \(\frac{a d c(a+b+c)}{a b+b c+c a}\)

Solution: Let \(\frac{a}{b}\) = \(\frac{b}{c}\) = \(\frac{c}{d}\) r (≠0)

⇒ a = br, b = cr, c = dr

⇒ ∴ b = dr², a = dr².

⇒ \(\frac{a b c(a+b+c)}{a b+b c+c a}=\frac{d r^3 \cdot d \cdot d r\left(d r^3+d r^2+d r\right)}{d r^3 \cdot d r^2+d r^2 \cdot d r+d r \cdot d r^3}\)

= \(\frac{d^3 r^4 \cdot d r\left(r^2+r+1\right)}{d^2 r^3\left(r^2+r+1\right)}\) = \(d^2 r^2=c^2\)

Example 7. If x: y = 5: 6 then find the value of \(\frac{3 x+4 y}{4 x+3 y}\)

Solution: \(\frac{x}{y}=\frac{5}{6}\)

⇒ Now, \(\frac{3 x+4 y}{4 x+3 y}=\frac{y\left(3 \frac{x}{y}+4\right)}{y\left(4 \frac{x}{y}+3\right)}\)

= \(\frac{5+8}{2} \times \frac{3}{19}=\frac{39}{38}\)

Example 8. What should be subtracted from each term of 4 : 9 to make the ratio equal to 8 : 7 = ?

Solution: Let the required term be x

⇒ \(\frac{4-x}{9-x}=\frac{8}{7}\)

⇒ or, 28 – 7x = 72 – 8x

⇒ or, x = 44

Example 9. If 7 + x, 11 + x, and 19 + x are in continued proportion, then find the value of x.

Solution: (7 + x) (19 + x) = (11 + x)²

⇒ or, 133 + 26x + x² = 121 + x² + 22x

⇒ or, 4x = – 12,

⇒ or, x = -3

The value of x = -3

Class 10 Maths Board Exam Solutions

Example 10. If \(\frac{x}{l m-n^2}=\frac{y}{m n-l^2}=\frac{z}{n l-m^2}\) then find the value of lx + my + xz.

Solution: Let \(\frac{x}{l m-n^2}=\frac{y}{m n-l^2}=\frac{z}{n l-m^2}=k(\neq 0)\)

∴ x = k (lm – n²); y = k(mn – l²); z = k(nl – m²)

∴ lx + my + xz

= lk (lm – n²) + mk (mn – l²) + nk (nl – m²)

= k (l²m- ln² + m²n – ml² + n²l – m²n)

= k.0 = 0.

The value of lx + my + xz = 0.