Variations: Class 10 Mathematics Notes, Short Questions and Numericals

Ratio and proportion, properties of proportion such as componendo and dividendo, and direct, inverse and joint variation with the constant of variation, shown through word problems.

Key points

A ratio compares two quantities of the same kind and is written a : b or a/b. The first term a is the antecedent and the second term b is the consequent. A ratio has no unit. Ratios such as 15 : 20 and 3 : 4 are equal, because both reduce to the same value.

A proportion is a statement that two ratios are equal. If a : b = c : d, we write a : b :: c : d. The terms a and d are the extremes and b and c are the means. In every proportion the product of extremes equals the product of means, so ad = bc.

Terms used with proportion:

Fourth proportional to a, b, c is the number d in a : b = c : d.
Third proportional to a and b is the number c in a : b = b : c.
Mean proportional between a and b is the number m in a : m = m : b, so m = √(ab).

Properties of proportion. If a/b = c/d, then:

Invertendo: b/a = d/c.
Alternendo: a/c = b/d.
Componendo: (a + b)/b = (c + d)/d.
Dividendo: (a − b)/b = (c − d)/d.
Componendo and dividendo: (a + b)/(a − b) = (c + d)/(c − d).

These properties are proved by adding or subtracting 1 on both sides of a/b = c/d. For example, a/b + 1 = c/d + 1 gives (a + b)/b = (c + d)/d, which is componendo. They are useful when an equation has a sum and a difference of the same terms on each side.

Variation describes how one quantity changes when another changes. If y varies directly as x, we write y ∝ x. This means y = kx, where k is a constant that does not change, called the constant of variation. The graph of y against x is a straight line through the origin. When x is doubled, y is also doubled.

If y varies inversely as x, we write y ∝ 1/x, which means y = k/x or xy = k. As x increases, y decreases so that the product stays the same. For example, if more workers are used on a job, the number of days needed falls.

Joint variation involves more than two quantities. If z varies jointly as x and y, then z ∝ xy and z = kxy. The area of a rectangle is jointly proportional to its length and width, with k = 1. A combination is also possible, for example z ∝ x/y means z = kx/y.

Steps for solving a variation problem:

Write the variation as a statement, for example y ∝ x.
Change it to an equation with a constant, y = kx.
Put the first set of given values in to find k.
Write the equation again with the value of k.
Put the new value of the known variable in and find the unknown.

Properties of variation help in changing between forms. If x ∝ y and y ∝ z, then x ∝ z. If x ∝ y and x ∝ z, then x ∝ yz is not true in general, but x² ∝ yz is. If x ∝ y, then x^n ∝ y^n for any power n. These are used mostly in short proofs and in problems with squares and square roots.

Study tip: In word problems, first decide whether the quantities rise together (direct) or one falls as the other rises (inverse). Then always follow the same steps: write the proportion, form the equation with k, find k from the first set of values and then solve. Show each step, because marks are given for the method.

Definitions and flashcards

Say the meaning first, then open the card to check.

Ratio

A comparison of two quantities of the same kind by division.

Proportion

A statement that two ratios are equal.

Mean proportional

The number m between a and b such that a : m = m : b.

Direct variation

A relation y = kx in which y increases as x increases.

Inverse variation

A relation y = k/x in which y decreases as x increases.

Joint variation

A relation in which one quantity varies as the product of two or more others.

Constant of variation

The fixed number k in the equation of the variation.

Componendo

The step that changes a/b = c/d into (a + b)/b = (c + d)/d.

Formula sheet

Name	Formula	What it means
Proportion	a/b = c/d ⇒ ad = bc	Product of extremes equals product of means.
Mean proportional	m = √(ab)	Mean proportional between a and b.
Direct variation	y = kx	k is the constant of variation.
Inverse variation	y = k/x, or xy = k	The product stays constant.
Joint variation	z = kxy	z varies jointly as x and y.
Componendo and dividendo	(a + b)/(a − b) = (c + d)/(c − d)	Applied when a/b = c/d.

Short questions with model answers

These are the kind of questions asked for 2, 3 and 5 marks. Try each one before opening the answer.

Define ratio and proportion. (2 marks)

A ratio is a comparison of two quantities of the same kind by division, written a : b. A proportion is an equality of two ratios, written a : b = c : d, for which ad = bc.

Find the mean proportional between 4 and 25. (2 marks)

Let the mean proportional be m. Then 4 : m = m : 25, so m² = 4 × 25 = 100. Hence m = 10.

What is meant by direct variation? Give the equation and one example. (3 marks)

When two quantities change in the same ratio, the first varies directly as the second. If y ∝ x, then y = kx, where k is a constant. For example, the cost of cloth is directly proportional to its length: if one metre costs Rs 200, then 5 m cost 5 × 200 = Rs 1,000.

Write the difference between direct and inverse variation. (3 marks)

In direct variation y = kx, so the ratio y/x stays constant and y rises when x rises. In inverse variation y = k/x, so the product xy stays constant and y falls when x rises. The graph of direct variation is a straight line through the origin, while inverse variation gives a curve.

If a/b = c/d, prove that (a + b)/(a − b) = (c + d)/(c − d). (5 marks)

Let a/b = c/d. Adding 1 to both sides gives (a + b)/b = (c + d)/d. Subtracting 1 from both sides gives (a − b)/b = (c − d)/d. Divide the first result by the second: [(a + b)/b] ÷ [(a − b)/b] = [(c + d)/d] ÷ [(c − d)/d]. The b and d cancel, leaving (a + b)/(a − b) = (c + d)/(c − d). This is what was to be proved.

The area of a rectangle varies jointly as its length and width. Write the equation and explain the constant. (3 marks)

If A is the area, l the length and w the width, then A ∝ lw, so A = klw. With the same units for all three, k = 1, which gives the well known formula A = lw. For example, l = 6 m and w = 4 m gives A = 24 m².

Numericals with worked steps

y varies directly as x. If y = 12 when x = 4, find y when x = 10.

y = kx.
12 = k × 4, so k = 3.
y = 3x.
When x = 10, y = 3 × 10 = 30.

Answer: y = 30

y varies inversely as x. If y = 6 when x = 8, find y when x = 12.

y = k/x, so k = xy = 8 × 6 = 48.
y = 48/x.
When x = 12, y = 48/12 = 4.

Answer: y = 4

z varies jointly as x and y. If z = 24 when x = 2 and y = 3, find z when x = 5 and y = 6.

z = kxy.
24 = k × 2 × 3 = 6k, so k = 4.
z = 4xy.
When x = 5 and y = 6, z = 4 × 5 × 6 = 120.

Answer: z = 120

Find x if (x + 3)/(x − 3) = 5/3, using componendo and dividendo.

Apply componendo and dividendo: ((x + 3) + (x − 3)) / ((x + 3) − (x − 3)) = (5 + 3)/(5 − 3).
Left side: 2x / 6 = x/3.
Right side: 8/2 = 4.
x/3 = 4, so x = 12.
Check: (12 + 3)/(12 − 3) = 15/9 = 5/3, which is correct.

Answer: x = 12

12 workers finish a job in 15 days. How many days will 20 workers take at the same rate?

Days vary inversely as the number of workers, so d = k/w.
k = 12 × 15 = 180.
For 20 workers, d = 180/20 = 9.

Answer: 9 days

Common mistakes

Treating an inverse variation as a direct one and dividing instead of multiplying. Check whether the product or the ratio is constant.
Forgetting to find the constant k first and putting the new values straight into the equation.
Applying componendo and dividendo without first checking that the equation has the form a/b = c/d.
Mixing up the means and the extremes, which gives the wrong cross product.
Writing y ∝ x as y = x, leaving out the constant k.

Practise this chapter

Practice questions and timed test Printable worksheet