Take a squared paper and arrange 48 counters on it in different number of rows as shown below. Number of rows R R 1 2 R 2 3 R 3 4 R 4 6 R 5 8 Number of columns C C 1 _ _ _ C 2 _ _ _ C 3 12 C 4 8 C 5 _ _ _ What do you observe? As R increases, C decreases. Is R 1 : R 2 = C 2 : C 1 ?

It can be observed that the number of rows will increase with a decrease in the number of columns, provided the number of squares are constant. Number of Rows R R 1 C 1 Number of Columns C R 2 C 2 The number of rows can be said to be in inversely proportion to the number of columns R ∝ 1 C , ⇒ R = k × 1 C , where k is a constant Then, R C = k (a constant) R 1 × C 1 = R 2 × C 2 = Number of squares Number of Rows (R) 2 3 4 6 8 Number of Columns (C) _____ _____ 12 8 _____ ( i ) R 1 × C 1 = 48 2 × C 1 = 48 C 1 = 48 2 = 24 ( i i ) R 2 × C 2 = 48 3 × C 2 = 48 C 2 = 48 3 = 16 ( v ) R 5 × C 5 = 48 8 × C 5 = 48 C 5 = 48 8 = 6 Now, we have the following table: Number of Rows (R) 2 3 4 6 8 Number of Columns (C) 24 16 12 8 6 From the table, we have R 1 = 2 , C 1 = 24 and R 2 = 3 , C 2 = 16 R 1 R 2 = 2 3 and C 2 C 1 = 16 24 = 2 3 Yes, R 1 R 2 = C 2 C 1 .

Take a squared paper and arrange 48 counters on it in different number of rows as shown below. Number of rows R R 1 2 R 2 3 R 3 4 R 4 6 R 5 8 Number of columns C C 1 _ _ _ C 2 _ _ _ C 3 12 C 4 8 C 5 _ _ _ What do you observe? As R increases, C decreases. Is R 3 : R 4 = C 4 : C 3 ?

It can be observed that the number of rows will increase with a decrease in the number of columns, provided the number of squares are constant. Number of Rows R R 1 C 1 Number of Columns C R 2 C 2 The number of rows can be said to be in inversely proportion to the number of columns when R ∝ 1 C , ⇒ R = k × 1 C , where k is a constant R C = k (a constant) R 1 × C 1 = R 2 × C 2 = Number of squares. Number of Rows (R) 2 3 4 6 8 Number of Columns (C) _____ _____ 12 8 _____ ( i ) R 1 × C 1 = 48 2 × C 1 = 48 C 1 = 48 2 = 24 ( i i ) R 2 × C 2 = 48 3 × C 2 = 48 C 2 = 48 3 = 16 ( v ) R 5 × C 5 = 48 8 × C 5 = 48 C 5 = 48 8 = 6 Now, we have the following table: Number of Rows (R) 2 3 4 6 8 Number of Columns (C) 24 16 12 8 6 From the table, we have R 3 = 4 , C 3 = 12 and R 4 = 6 , C 4 = 8 R 3 R 4 = 4 6 = 2 3 and C 4 C 3 = 8 12 = 2 3 Yes, R 3 R 4 = C 4 C 3 .

Take a squared paper and arrange 48 counters on it in different number of rows as shown below. Number of rows R R 1 2 R 2 3 R 3 4 R 4 6 R 5 8 Number of columns C C 1 _ _ _ C 2 _ _ _ C 3 12 C 4 8 C 5 _ _ _ What do you observe? As R increases, C decreases. Is R and C inversely proportional to each other.

Number of Rows (R) 2 3 4 6 8 Number of Columns (C) _____ _____ 12 8 _____ ( i ) R 1 × C 1 = 48 2 × C 1 = 48 ⇒ C 1 = 48 2 = 24 ( i i ) R 2 × C 2 = 48 3 × C 2 = 48 ⇒ C 2 = 48 3 = 16 ( v ) R 5 × C 5 = 48 8 × C 5 = 48 ⇒ C 5 = 48 8 = 6 Now, we have the following table: Number of Rows (R) 2 3 4 6 8 Number of Columns (C) 24 16 12 8 6 From the table it can be observed that with the increase in the number of rows there is a decrease in the number of columns. We know that, two variables R and C are said to be in inverse proportion if R C is a constant. Now, in the above given table; R 1 C 1 = 2 × 24 = 48 R 2 C 2 = 3 × 16 = 48 R 3 C 3 = 4 × 12 = 48 R 4 C 4 = 6 × 8 = 48 R 5 C 5 = 8 × 6 = 48 We can see that, R 1 C 1 = R 2 C 2 = R 3 C 3 = R 4 C 4 = R 5 C 5 Hence, R and C are inversely proportional to each other.

Take a squared paper and arrange 48 counters on it in different number of rows as shown below. Number of rows R R 1 2 R 2 3 R 3 4 R 4 6 R 5 8 Number of columns C C 1 24 C 2 16 C 3 12 C 4 8 C 5 6 What do you observe? As R increases, C decreases. Do this activity with 36 squares.

Number of dots on a squared paper = 36 It can be observed that the number of rows will increase with a decrease in the number of columns, provided the number of squares are constant. The number of rows can be said to be in inversely proportion to the number of columns Number of rows × Number of columns = Number of squares. ( i ) R 1 × C 1 = 36 2 × C 1 = 36 C 1 = 36 2 = 18 ( i i ) R 2 × C 2 = 36 3 × C 2 = 36 C 2 = 36 3 = 12 ( i i i ) R 3 × C 3 = 36 4 × C 3 = 36 C 3 = 36 4 = 9 ( i v ) R 4 × C 4 = 36 6 × C 4 = 36 C 3 = 36 6 = 6 ( v ) R 5 × C 5 = 36 8 × C 5 = 36 C 5 = 36 8 = 4 . 5 Thus, with 8 rows number of columns cannot be determined with complete dots in a last column. Now, we have the following table, when the number of squares is 36 . Number of Rows (R) 2 3 4 6 8 Number of Columns (C) 18 12 9 6 __

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The average age of consisting doctors and lawyers is $40$ . If the doctors average age is $35$ and the lawyers average age is $50$ , find the ratio of the number of doctors to the number of lawyers.

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Important Questions on Direct and Inverse Proportions

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IMPORTANT

Mathematics Class 8>Chapter 10 - Direct and Inverse Proportions>Exercise 2>Q 2

The following ingredients are required to make Halwa for

5

persons: Suji/Rawa

= 250 g

, Sugar

= 300 g

, Ghee

= 200 g

, Water

= 500 ml

. Using the concept of proportion, estimate the changes in the quantity of ingredients to prepare for a class of

30

students.

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IMPORTANT

Mathematics Class 8>Chapter 10 - Direct and Inverse Proportions>Exercise 3>Q (i)

Observe the following table and find which pair of variables ( $x$ and $y$ ) are in inverse proportion

$x$	$50$	$40$	$30$	$20$
$y$	$5$	$6$	$7$	$8$

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8th Telangana Board

IMPORTANT

Mathematics Class 8>Chapter 10 - Direct and Inverse Proportions>Exercise 3>Q (ii)

Observe the following table and find which pair of variables ( $x$ and $y$ ) are in inverse proportion

$x$	$100$	$200$	$300$	$400$
$y$	$60$	$30$	$20$	$15$

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8th Telangana Board

IMPORTANT

Mathematics Class 8>Chapter 10 - Direct and Inverse Proportions>Exercise 3>Q (iii)

Observe the following table and find which pair of variables ( $x$ and $y$ ) are in inverse proportion

$x$	$90$	$60$	$45$	$30$	$20$	5
$y$	$10$	$15$	$20$	$25$	$30$	$25$

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8th Telangana Board

IMPORTANT

Mathematics Class 8>Chapter 10 - Direct and Inverse Proportions>Exercise 3>Q 3

Take a squared paper and arrange $48$ counters on it in different number of rows as shown below.

Question Image

Number of rows $(R)$	$(R_{1})$ $2$	$(R_{2}) 3$	$(R_{3}) 4$	$(R_{4}) 6$	$(R_{5}) 8$
Number of columns $(C)$	$(C_{1})___$	$(C_{2})___$	$(C_{3}) 12$	$(C_{4}) 8$	$(C_{5})___$

What do you observe? As R increases, C decreases.

Is $R_{1} : R_{2} = C_{2} : C_{1}$ ?

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8th Telangana Board

IMPORTANT

Mathematics Class 8>Chapter 10 - Direct and Inverse Proportions>Exercise 3>Q 3

Take a squared paper and arrange $48$ counters on it in different number of rows as shown below.

Question Image

Number of rows $(R)$	$(R_{1})$ $2$	$(R_{2}) 3$	$(R_{3}) 4$	$(R_{4}) 6$	$(R_{5}) 8$
Number of columns $(C)$	$(C_{1})___$	$(C_{2})___$	$(C_{3}) 12$	$(C_{4}) 8$	$(C_{5})___$

What do you observe? As R increases, C decreases.

Is $R_{3} : R_{4} = C_{4} : C_{3}$ ?

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8th Telangana Board

IMPORTANT

Mathematics Class 8>Chapter 10 - Direct and Inverse Proportions>Exercise 3>Q 3

Take a squared paper and arrange $48$ counters on it in different number of rows as shown below.

Question Image

Number of rows $(R)$	$(R_{1})$ $2$	$(R_{2}) 3$	$(R_{3}) 4$	$(R_{4}) 6$	$(R_{5}) 8$
Number of columns $(C)$	$(C_{1})___$	$(C_{2})___$	$(C_{3}) 12$	$(C_{4}) 8$	$(C_{5})___$

What do you observe? As R increases, C decreases.

Is $R$ and $C$ inversely proportional to each other.

HARD

8th Telangana Board

IMPORTANT

Mathematics Class 8>Chapter 10 - Direct and Inverse Proportions>Exercise 3>Q 3

Take a squared paper and arrange $48$ counters on it in different number of rows as shown below.

Question Image

Number of rows $(R)$	$(R_{1})$ $2$	$(R_{2}) 3$	$(R_{3}) 4$	$(R_{4}) 6$	$(R_{5}) 8$
Number of columns $(C)$	$(C_{1}) 24$	$(C_{2}) 16$	$(C_{3}) 12$	$(C_{4}) 8$	$(C_{5}) 6$

What do you observe? As R increases, C decreases.

Do this activity with $36$ squares.

The average age of consisting doctors and lawyers is 40. If the doctors average age is 35 and the lawyers average age is 50, find the ratio of the number of doctors to the number of lawyers.

Important Questions on Direct and Inverse Proportions

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The average age of consisting doctors and lawyers is $40$ . If the doctors average age is $35$ and the lawyers average age is $50$ , find the ratio of the number of doctors to the number of lawyers.