Maharashtra Board Class 8 Maths part 1 Chapter 7 Variation PDF Download

Variation

Let's Recall

If the rate of notebooks is ₹ 240 per dozen, what is the cost of 3 notebooks?

Also find the cost of 9 notebooks; 24 notebooks and 50 notebooks and complete the following table.

From the above table we see that the ratio of number of notebooks (x) and their cost (y) in each pair is \(\frac{1}{20}\). It is constant. The number of notebooks and their cost are in the same proportion. In such a case, if one number increases then the other number increases in the same proportion.

Let's Learn

Direct Variation

The statement 'x and y are in the same proportion' can be written as 'x and y are in direct variation' or 'there is a direct variation between x and y'. Using mathematical symbol it can be written as \(x \propto y\). [∝ (alpha) is a greek letter, used to denote variation.]

\(x \propto y\) is written in the form of equation as \(x = ky\), where k is a constant.

\(x = ky\) or \(\frac{x}{y} = k\) is the equation form of direct variation where k is the constant of variation.

Observe how the following statements are written using the symbol of variation.

(i) Area of a circle is directly proportional to the square of its radius.

If the area of a circle = A, its radius = r, the above statement is written as \(A \propto r^2\).

(ii) Pressure of a liquid (p) varies directly as the depth (d) of the liquid; this statement is written as \(p \propto d\).

To understand the method of symbolic representation of direct variation, study the following examples.

Ex. (1) x varies directly as y, when x = 5, y = 30. Find the constant of variation and equation of variation.

Solution: x varies directly as y, that is as \(x \propto y\)

\(x = ky\) ......... k is constant of variation.

when x = 5, y = 30, is given

\(5 = k \times 30\) \(\therefore k = \frac{1}{6}\) (constant of variation)

\(\therefore\) equation of variation is \(x = ky\), that is \(x = \frac{y}{6}\) or \(y = 6x\)

Teacher's Note

Direct variation means when one thing increases, the other also increases. Like when you buy more notebooks, the cost also increases.

Exam Trick

Remember: Direct variation means both things increase together. If notebooks increase, cost increases. Both go up together like friends walking together.

Points to Remember

Direct variation means when one number increases, the other also increases in the same amount.

The symbol \(\propto\) means "is proportional to" or "varies directly as".

In the equation \(x = ky\), k is the constant of variation and it never changes.

To find k, divide one number by the other: \(k = \frac{x}{y}\)

Ex. (2) Cost of groundnuts is directly proportional to its weight. If cost of 5 kg groundnuts is ₹ 450 then find the cost of 1 quintal groundnuts. (1 quintal = 100 kg)

Solution: Let the cost of groundnuts be x and weight of groundnuts be y.

It is given that x varies directly as y \(\therefore x \propto y\) or \(x = ky\)

It is given that when x = 450 then y = 5, hence we will find k.

\(x = ky\) \(\therefore 450 = 5k\) \(\therefore k = 90\) (constant of variation)

\(\therefore\) equation of variation is \(x = 90y\).

\(\therefore\) if y = 100, \(x = 90 \times 100 = 9000\)

\(\therefore\) cost of 1 quintal groundnut is ₹ 9000.

Teacher's Note

This is like shopping at the market. If 5 kg of groundnuts cost ₹ 450, then 100 kg will cost much more. The price increases with weight.

Exam Trick

Remember: First find k by dividing the given numbers. Then use k to find any other value. It is like finding the price per kg first.

Points to Remember

Always find the constant k first using the given values.

Once you have k, you can find any other value using \(x = ky\).

Check: The ratio \(\frac{x}{y}\) must always be the same (equal to k).

In real life, direct variation is like: more apples = more cost, more workers = more work done.

Practice Set 7.1

1. Write the following statements using the symbol of variation.

(1) Circumference (c) of a circle is directly proportional to its radius (r).

(2) Consumption of petrol (l) in a car and distance travelled by that car (d) are in direct variation.

2. Complete the following table considering that the cost of apples and their number are in direct variation.

3. If \(m \propto n\) and when m = 154, n = 7. Find the value of m, when n = 14

4. If n varies directly as m, complete the following table.

5. y varies directly as square root of x. When x = 16, y = 24. Find the constant of variation and equation of variation.

6. The total remuneration paid to labourers, employed to harvest soyabeen is in direct variation with the number of labourers. If remuneration of 4 labourers is ₹ 1000, find the remuneration of 17 labourers.

Let's Recall

The following table shows the number of rows and number of students in each row when they are made to stand for drill.

From the table we observe that the product of number of students in each row and total number of rows in each pair is 240; which is constant. It means, number of students in a row and number of rows are in inverse proportion.

In a pair of numbers, if the increase in one number causes decrease in the other number in the same proportion, the pair is in inverse variation. In such an example, if one number of the pair is doubled, the other is halved.

Let's Learn

Inverse Variation

The statement 'x is inversely proportional to y' can also be expressed as 'there is inverse variation in x and y.' If x and y are in inverse proportion, \(x \times y\) is constant. Assuming the constant to be k, it is easy to solve a problem.

If x varies inversely as y then \(x \times y\) is constant.

'x inversely varies as y' is written as \(x \propto \frac{1}{y}\).

If \(x \propto \frac{1}{y}\) then \(x = \frac{k}{y}\) or \(x \times y = k\); this is the equation of variation. k is the constant of variation.

Solved Examples

Ex. (1) If a varies inversely as b then complete the following table.

Solution: (i) \(a \propto \frac{1}{b}\) that is \(a \times b = k\)

when a = 6, b = 20 \(\therefore k = 6 \times 20 = 120\) (constant of variation)

(ii) If a = 12, b = ?

\(a \times b = 120\)

\(\therefore 12 \times b = 120\)

\(\therefore b = 10\)

(iii) If a = 15, b = ?

\(a \times b = 120\)

\(\therefore 15 \times b = 120\)

\(\therefore b = 8\)

(iv) If b = 4, a = ?

\(a \times b = 120\)

\(\therefore a \times 4 = 120\)

\(\therefore a = 30\)

Teacher's Note

Inverse variation is like a seesaw. When one side goes up, the other goes down. If more students stand in a row, fewer rows are needed.

Exam Trick

Remember: Inverse variation means multiply. Use \(x \times y = k\) always. If x increases, y must decrease to keep the product the same.

Points to Remember

In inverse variation, when one thing increases, the other decreases.

The product of the two things always stays the same: \(x \times y = k\)

Use the given values to find k first, then find any missing value.

Real life examples: more workers finish work in fewer days; faster speed takes less time.

Ex. (2) \(f \propto \frac{1}{d^2}\), when d = 5, f = 18

Hence, (i) if d = 10 find f. (ii) when f = 50 find d.

Solution: \(f \propto \frac{1}{d^2}\) \(\therefore f \times d^2 = k\), when d = 5 and f = 18.

\(\therefore 18 \times 5^2 = k\) \(\therefore k = 18 \times 25 = 450\) (constant of variation)

(i) if d = 10 then f = ?

\(f \times d^2 = 450\)

\(\therefore f \times 10^2 = 450\)

\(\therefore f \times 100 = 450\)

\(\therefore f = 4.5\)

(ii) if f = 50, then d = ?

\(f \times d^2 = 450\)

\(\therefore 50 \times d^2 = 450\)

\(\therefore d^2 = 9\)

\(\therefore d = 3\) or \(d = -3\)

Teacher's Note

Sometimes the variation is with a square or square root. Like light gets weaker with the square of distance from a bulb.

Exam Trick

Remember: If you see \(x^2\) or \(\sqrt{x}\), include it in the equation. Multiply or divide carefully with these powers.

Points to Remember

Sometimes variation involves squares: \(x \times y^2 = k\) or \(x \times \sqrt{y} = k\)

Always check if there is a power or square root in the equation.

Write the correct equation first, then find k, then find the answer.

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