Proportional Reasoning-1

Image A has a width of 60 mm and a height of 40 mm, while image D has a width of 90 mm and a height of 60 mm. By what factors do the width and height of image D change as compared to image A? Are the factors the same?

Answer

Given:

Image A — width = 60 mm, height = 40 mm

Image D — width = 90 mm, height = 60 mm

Factor of change in width = $\dfrac{\text{width of D}}{\text{width of A}} = \dfrac{90}{60} = \dfrac{3}{2}$

Factor of change in height = $\dfrac{\text{height of D}}{\text{height of A}} = \dfrac{60}{40} = \dfrac{3}{2}$

Both the width and the height have changed by the same factor $\dfrac{3}{2}$.

Since the width and height changed by the same factor, images A and D are similar because their corresponding dimensions are proportional.

Hence, the width and height both change by a factor of $\dfrac{3}{2}$, and the factors are the same.

By what factor should we multiply the ratio 60 : 40 (image A) to get 90 : 60 (image D)?

Answer

Given ratio of image A = 60 : 40

Required ratio of image D = 90 : 60

Find the factor by dividing the corresponding terms:

First term: $60 \times f = 90 \Rightarrow f = \dfrac{90}{60} = \dfrac{3}{2}$

Check second term: $40 \times \dfrac{3}{2} = 60$ $\quad$[matches image D]

So, multiplying both terms of 60 : 40 by $\dfrac{3}{2}$ gives 90 : 60.

Hence, we should multiply both terms of 60 : 40 by $\dfrac{3}{2}$ to get 90 : 60.

Are the ratios 3 : 4 and 72 : 96 proportional?

Answer

The ratio 3 : 4 is already in its simplest form.

To reduce 72 : 96, divide both terms by their HCF.

Let us find HCF of 72 and 96:

$\begin{array}{r} 1 \\ 72 \overline{) 96} \\ \underline{-72} \\ 24 \overline{) 72} ( 3 \\ \underline{-72} \\ 0 \end{array}$

HCF = 24

$∴ 72 : 96 = \dfrac{72 \div 24}{96 \div 24} = \dfrac{3}{4} = 3 : 4$

Both ratios in their simplest form are the same, i.e., 3 : 4.

Hence, the ratios 3 : 4 and 72 : 96 are proportional.

Kesang wanted to make lemonade for a celebration. She made 6 glasses of lemonade in a vessel and added 10 spoons of sugar to the drink. Her father expected more people to join the celebration. So he asked her to make 18 more glasses of lemonade. To make the lemonade with the same sweetness, how many spoons of sugar should she add?

Answer

Given:

For 6 glasses of lemonade, sugar added = 10 spoons.

Kesang has to make 18 additional glasses. To maintain the same sweetness, the ratio of the number of glasses of lemonade to the number of spoons of sugar must remain proportional.

We can model the additional quantity as:

6 : 10 :: 18 : x

Find the factor of change in the first term: $\dfrac{18}{6} = 3$

The second term must change by the same factor:

10 x 3 = 30

So, 6 : 10 :: 18 : 30.

Thus, she must add 30 spoons of sugar for the additional 18 glasses. She will then have 24 glasses of lemonade and 40 spoons of sugar in total.

Hence, she should add 30 more spoons of sugar for the additional 18 glasses of lemonade.

Nitin and Hari were constructing a compound wall around their house. Nitin was building the longer side, 60 ft in length, and Hari was building the shorter side, 40 ft in length. Nitin used 3 bags of cement but Hari used only 2 bags of cement. Nitin was worried that the wall Hari built would not be as strong as the wall he built because she used less cement.

Is Nitin correct in his thinking?

Answer

To compare the use of cement, we compare the ratio of the length of the wall to the number of bags of cement used by each person.

The ratio in Nitin's case = 60 : 3

$60 : 3 = \dfrac{60 \div 3}{3 \div 3} = 20 : 1$ $\quad$[HCF of 60 and 3 is 3]

The ratio in Hari's case = 40 : 2

$40 : 2 = \dfrac{40 \div 2}{2 \div 2} = 20 : 1$ $\quad$[HCF. of 40 and 2 is 2]

Both ratios are the same in their simplest form (20 : 1), so they are proportional. This means the same length of wall uses the same amount of cement in both cases, and the walls are equally strong.

Hence, Nitin is not correct because both Nitin and Hari used cement in the same proportion.

When Neelima was 3 years old, her mother's age was 10 times her age. What is the ratio of Neelima's age to her mother's age? What would be the ratio of their ages when Neelima is 12 years old? Would it remain the same?

Answer

When Neelima was 3 years old, her mother's age was:

10 × 3 = 30 years

Ratio of Neelima's age to her mother's age = 3 : 30

$3 : 30 = \dfrac{3 \div 3}{30 \div 3} = 1 : 10$ $\quad$[HCF of 3 and 30 is 3]

When Neelima is 12 years old, the time passed = 12 – 3 = 9 years.

So, her mother's age = 30 + 9 = 39 years.

Ratio of their ages now = 12 : 39

$12 : 39 = \dfrac{12 \div 3}{39 \div 3} = 4 : 13$ $\quad$[HCF of 12 and 39 is 3]

Comparing the two ratios:

1 : 10 ≠ 4 : 13

So, the ratio does not remain the same.

Hence, the ratio is 1 : 10 when Neelima is 3 years old and 4 : 13 when she is 12 years old; the ratio does not remain the same.

Fill in the missing numbers for the following ratios that are proportional to 14 : 21.

(i) .......... : 42
(ii) 6 : ..........
(iii) 2 : ..........

Answer

The given ratio is 14 : 21.

(i) .......... : 42

The second term has changed from 21 to 42.

Factor of change = $\dfrac{42}{21} = 2$

Therefore, the first term must also be multiplied by 2:

14 x 2 = 28

So, the ratio is 28 : 42.

(ii) 6 : ..........

The first term has changed from 14 to 6.

Factor of change = $\dfrac{6}{14} = \dfrac{3}{7}$

Therefore, the second term must also be multiplied by $\dfrac{3}{7}$:

$21 \times \dfrac{3}{7} = 9$

So, the ratio is 6 : 9.

(iii) 2 : ..........

The first term has changed from 14 to 2.

Dividing 14 by its HCF. with 21: $14 \div 7 = 2$ $\quad$[HCF. of 14 and 21 is 7]

So the second term must also be divided by 7:

$21 \div 7 = 3$

So, the ratio is 2 : 3.

Filter coffee is made by mixing coffee decoction with milk. Manjunath usually mixes 15 mL of coffee decoction with 35 mL of milk to make one cup of filter coffee in his coffee shop. If customers want ‘stronger’ filter coffee, Manjunath mixes 20 mL of decoction with 30 mL of milk. The ratio here is 20 : 30. Why is this coffee stronger? For a 'lighter' filter coffee, he mixes 10 mL of decoction with 40 mL of milk (ratio 10 : 40). Why is this coffee lighter?

Answer

In each cup the total quantity of coffee is the same: 15 + 35 = 20 + 30 = 10 + 40 = 50 mL.

So the strength of the coffee depends on how much of this 50 mL is decoction.

Regular coffee (decoction : milk = 15 : 35)

Fraction of decoction in the cup = $\dfrac{15}{50} = \dfrac{3}{10}$

Simplest form of the ratio =

$15 : 35 = \dfrac{15 \div 5}{35 \div 5} = 3 : 7$ $\quad$[HCF of 15 and 35 is 5]

Stronger coffee (decoction : milk = 20 : 30)

Fraction of decoction in the cup = $\dfrac{20}{50} = \dfrac{2}{5}$

Simplest form of the ratio =

$20 : 30 = \dfrac{20 \div 10}{30 \div 10} = 2 : 3$ $\quad$[HCF of 20 and 30 is 10]

Since $\dfrac{2}{5} \gt \dfrac{3}{10}$, this cup has a larger share of decoction (the ratio of decoction to milk has increased from 3 : 7 to 2 : 3). More decoction in the same amount of coffee makes the taste stronger.

Hence, the 20 : 30 coffee is stronger because it contains a greater proportion of decoction.

Lighter coffee (decoction : milk = 10 : 40)

Fraction of decoction in the cup = $\dfrac{10}{50} = \dfrac{1}{5}$

Simplest form of the ratio =

$10 : 40 = \dfrac{10 \div 10}{40 \div 10} = 1 : 4$ $\quad$[HCF of 10 and 40 is 10]

Since $\dfrac{1}{5} \lt \dfrac{3}{10}$, this cup has a smaller share of decoction (the ratio of decoction to milk has decreased from 3 : 7 to 1 : 4). Less decoction in the same amount of coffee makes the taste lighter.

Hence, the 10 : 40 coffee is lighter because it contains a smaller proportion of decoction.

The following table shows the different ratios in which Manjunath mixes coffee decoction with milk. Write in the last column whether the coffee is stronger or lighter than the regular coffee.

Answer

The regular coffee has the ratio decoction : milk = 15 : 35

HCF of 15 and 35 = 5

Simplest form of the ratio =

$15 : 35 = \dfrac{15 \div 5}{35 \div 5} = 3 : 7$

So, the regular coffee has ratio 3 : 7.

Now simplify each ratio and compare it with 3 : 7.

A higher proportion of decoction means the coffee is stronger, a lower proportion means it is lighter, and the same proportion means it is regular.

To compare a ratio a : b with 3 : 7, we cross multiply: if 7a > 3b it is stronger, if 7a < 3b it is lighter, and if 7a = 3b it is regular.

300 : 600 = 1 : 2 ⇒ 7 × 300 = 2100 and 3 × 600 = 1800 ⇒ 2100 > 1800 ⇒ Strong

150 : 500 = 3 : 10 ⇒ 7 × 150 = 1050 and 3 × 500 = 1500 ⇒ 1050 < 1500 ⇒ Light

200 : 400 = 1 : 2 ⇒ 7 × 200 = 1400 and 3 × 400 = 1200 ⇒ 1400 > 1200 ⇒ Strong

24 : 56 = 3 : 7 ⇒ 7 × 24 = 168 and 3 × 56 = 168 ⇒ 168 = 168 ⇒ Regular

100 : 300 = 1 : 3 ⇒ 7 × 100 = 700 and 3 × 300 = 900 ⇒ 700 < 900 ⇒ Light

The completed table is:

Circle the following statements of proportion that are true.

(i) 4 : 7 :: 12 : 21
(ii) 8 : 3 :: 24 : 6
(iii) 7 : 12 :: 12 : 7
(iv) 21 : 6 :: 35 : 10
(v) 12 : 18 :: 28 : 12
(vi) 24 : 8 :: 9 : 3

Answer

Two ratios are proportional if they are equal when written in their simplest form.

(i) 4 : 7 :: 12 : 21

The ratio 4 : 7 is already in its simplest form.

Simplest form of 12 : 21 =

$12 : 21 = \dfrac{12 \div 3}{21 \div 3} = 4 : 7$ $\quad$[HCF of 12 and 21 is 3]

So, 4 : 7 = 12 : 21.

The statement of proportion is true.

(ii) 8 : 3 :: 24 : 6

The ratio 8 : 3 is already in its simplest form.

Simplest form of 24 : 6 =

$24 : 6 = \dfrac{24 \div 6}{6 \div 6} = 4 : 1$ $\quad$[HCF of 24 and 6 is 6]

So, 8 : 3 ≠ 24 : 6.

The statement of proportion is false.

(iii) 7 : 12 :: 12 : 7

The ratio 7 : 12 is already in its simplest form.

The ratio 12 : 7 is already in its simplest form.

So, 7 : 12 ≠ 12 : 7.

The statement of proportion is false.

(iv) 21 : 6 :: 35 : 10

Simplest form of 21 : 6 =

$21 : 6 = \dfrac{21 \div 3}{6 \div 3} = 7 : 2$ $\quad$[HCF of 21 and 6 is 3]

Simplest form of 35 : 10 =

$35 : 10 = \dfrac{35 \div 5}{10 \div 5} = 7 : 2$ $\quad$[HCF of 35 and 10 is 5]

So, 21 : 6 = 35 : 10.

The statement of proportion is true.

(v) 12 : 18 :: 28 : 12

Simplest form of 12 : 18 =

$12 : 18 = \dfrac{12 \div 6}{18 \div 6} = 2 : 3$ $\quad$[HCF of 12 and 18 is 6]

Simplest form of 28 : 12 =

$28 : 12 = \dfrac{28 \div 4}{12 \div 4} = 7 : 3$ $\quad$[HCF of 28 and 12 is 4]

So, 12 : 18 ≠ 28 : 12.

The statement of proportion is false.

(vi) 24 : 8 :: 9 : 3

Simplest form of 24 : 8 =

$24 : 8 = \dfrac{24 \div 8}{8 \div 8} = 3 : 1$ $\quad$[HCF of 24 and 8 is 8]

Simplest form of 9 : 3 =

$9 : 3 = \dfrac{9 \div 3}{3 \div 3} = 3 : 1$ $\quad$[HCF of 9 and 3 is 3]

So, 24 : 8 = 9 : 3.

The statement of proportion is true.

Give 3 ratios that are proportional to 4 : 9.

Answer

The given ratio is 4 : 9.

To get proportional ratios, multiply both terms by the same number.

Multiplying by 2: $\dfrac{4 \times 2}{9 \times 2} = \dfrac{8}{18} = 8 : 18$

Multiplying by 3: $\dfrac{4 \times 3}{9 \times 3} = \dfrac{12}{27} = 12 : 27$

Multiplying by 4: $\dfrac{4 \times 4}{9 \times 4} = \dfrac{16}{36} = 16 : 36$

Hence, three ratios proportional to 4 : 9 are 8 : 18, 12 : 27 and 16 : 36. (Other answers are also possible.)

Fill in the missing numbers for these ratios that are proportional to 18 : 24.

3 : .......... 12 : .......... 20 : .......... 27 : ..........

Answer

First reduce 18 : 24 to its simplest form.

HCF of 18 and 24 is 6.

$18 : 24 = \dfrac{18 \div 6}{24 \div 6} = \dfrac{3}{4} = 3 : 4$

So every proportional ratio must reduce to 3 : 4, i.e. the second term is always $\dfrac{4}{3}$ times the first term.

3 : .......... ⇒ second term = $\dfrac{4}{3} \times 3 = 4$ ⇒ 3 : 4

12 : .......... ⇒ second term = $\dfrac{4}{3} \times 12 = 16$ ⇒ 12 : 16

20 : .......... ⇒ second term = $\dfrac{4}{3} \times 20 = \dfrac{80}{3}$ ⇒ $20 : \dfrac{80}{3}$

27 : .......... ⇒ second term = $\dfrac{4}{3} \times 27 = 36$ ⇒ 27 : 36

Hence, the completed ratios are 3 : 4, 12 : 16, $20 : \dfrac{80}{3}$ and 27 : 36.

Look at the following rectangles. Which rectangles are similar to each other? You can verify this by measuring the width and height using a scale and comparing their ratios.

Answer

Two rectangles are similar when the ratio of their corresponding sides (width : height) is the same in simplest form.

To check which rectangles are similar, follow these steps:

Step 1: Using a scale, measure the width and height of each rectangle A, B, C, D and E.

Step 2: Write the ratio width : height for each rectangle.

Step 3: Reduce each ratio to its simplest form.

Step 4: The rectangles whose simplest-form ratios are equal are similar to one another.

Hence, on measuring and comparing the ratios of width to height, the rectangles having the same simplest-form ratio are similar to each other.

Look at the following rectangle. Can you draw a smaller rectangle and a bigger rectangle with the same width to height ratio in your notebooks? Compare your rectangles with your classmates' drawings. Are all of them the same? If they are different from yours, can you think why? Are they wrong?

Answer

Yes, we can draw a smaller rectangle and a bigger rectangle with the same width to height ratio as the given rectangle.

To get a smaller rectangle, divide both the width and the height by the same number. To get a bigger rectangle, multiply both the width and the height by the same number.

When we compare our rectangles with those of our classmates, they may not all be the same size. Different students may choose different numbers to multiply or divide by.

However, if the width to height ratio remains the same, all such rectangles are proportional to the given rectangle.

Hence, the rectangles drawn by different students may be of different sizes, but they are not wrong as long as they have the same width to height ratio as the given rectangle.

The following figure shows a small portion of a long brick wall with patterns made using coloured bricks. Each wall continues this pattern throughout the wall. What is the ratio of grey bricks to coloured bricks? Try to give the ratios in their simplest form.

Answer

Since the pattern repeats throughout the wall, we count the grey and coloured bricks in one repeating block of the pattern.

(a)

Number of grey bricks in one block = 2 + 3 + 4 = 9

Number of coloured bricks in one block = 3 + 2 + 1 = 6

Ratio of grey bricks to coloured bricks = 9 : 6

HCF of 9 and 6 is 3.

$9 : 6 = \dfrac{9 \div 3}{6 \div 3} = \dfrac{3}{2} = 3 : 2$

Hence, the ratio of grey bricks to coloured bricks is 3 : 2.

(b)

Number of grey bricks in one block = 16

Number of coloured bricks in one block = 12

Ratio of grey bricks to coloured bricks = 16 : 12

HCF of 16 and 12 is 4.

$16 : 12 = \dfrac{16 \div 4}{12 \div 4} = \dfrac{4}{3} = 4 : 3$

Hence, the ratio of grey bricks to coloured bricks is 4 : 3

For the mid-day meal in a school with 120 students, the cook usually makes 15 kg of rice. On a rainy day, only 80 students came to school. How many kilograms of rice should the cook make so that the food is not wasted?

Answer

Given:

For 120 students, rice made = 15 kg

For 80 students, rice made = x kg

The number of students and the amount of rice should be proportional.

So, 120 : 15 :: 80 : x

Find the factor of change in the first term by dividing the terms:

$\dfrac{80}{120} = \dfrac{2}{3}$

The number of students is reduced by a factor of $\dfrac{2}{3}$.

On multiplying the weight of rice by the same factor, we get:

$15 \times \dfrac{2}{3} = 10$ kg

Hence, the cook should make 10 kg of rice on that day.

A car travels 90 km in 150 minutes. If it continues at the same speed, what distance will it cover in 4 hours?

Answer

Given:

Distance covered in 150 minutes = 90 km

Time = 4 hours

If the car continues at the same speed, the ratio of the time taken should be proportional to the ratio of the distance covered.

First, the units of time in both ratios must be the same.

Since 150 is in minutes, we convert 4 hours into minutes:

4 hours = 4 × 60 = 240 minutes

So, the proportion is set up as:

150 : 90 :: 240 : x

By cross multiplication, we get:

150 × x = 240 × 90

$x = \dfrac{240 \times 90}{150} \\[1em] x = \dfrac{21600}{150} \\[1em] x = 144$

Hence, the distance covered by the car in 4 hours is 144 km.

A small farmer in Himachal Pradesh sells each 200 g packet of tea for ₹200. A large estate in Meghalaya sells each 1 kg packet of tea for ₹800. Are the weight-to-price ratios in both places proportional? Which tea is more expensive?

Answer

Given:

Himachal Pradesh: 200 g of tea for ₹200

Meghalaya: 1 kg of tea for ₹800

To compare the ratios, the weights must be in the same unit.

Converting 1 kg into grams:

1 kg = 1000 g

The weight-to-price ratio of the Himachal tea = 200 : 200

In its simplest form: 200 : 200 = 1 : 1

The weight-to-price ratio of the Meghalaya tea = 1000 : 800

In its simplest form: 1000 : 800 = 5 : 4

Since 1 : 1 and 5 : 4 are not the same in their simplest forms, the ratios are not proportional.

Now, to find which tea is more expensive, compare the price of tea for the same weight (1 kg) in both places.

Price of 1 kg of tea in Meghalaya = ₹800

In Himachal, 200 g costs ₹200.

Since 1 kg = 1000 g = 5 × 200 g, the cost of 1 kg is:

5 × ₹200 = ₹1000

So, 1 kg of tea costs ₹1000 in Himachal Pradesh and ₹800 in Meghalaya.

Hence, the ratios are not proportional, and the tea from Himachal Pradesh is more expensive.

The Earth travels approximately 940 million kilometres around the Sun in a year. How many kilometres will it travel in a week?

Answer

Given:

Distance travelled by the Earth in 1 year = 940 million km

We know that 1 year = 52 weeks.

We know that 1 million = 10 lakh

So 940 million km = 940,000,000 km

Let the distance travelled in 1 week be x km.

The ratio of the time taken should be proportional to the ratio of the distance covered.

So, 52 : 940000000 :: 1 : x

By cross multiplication, we get:

52 × x = 940000000 × 1

⇒ x = $\dfrac{940000000}{52}$

⇒ x = 18076923

Hence, the Earth travels approximately 18076923 km in a week.

A mason is building a house in the shape shown in the diagram. He needs to construct both the outer walls and the inner wall that separates two rooms. To build a wall of 10-feet, he requires approximately 1450 bricks. How many bricks would he need to build the house? Assume all walls are of the same height and thickness.

Answer

Given:

For a wall of 10 feet, bricks required = 1450

From the diagram, find the total length of all the walls (outer walls + inner wall).

Now total length of walls = 12 + 12 + 12 + 15 + 9 + 15 + 9 + 9 + 9 + 6

= 108 ft

The ratio of the length of the wall to the number of bricks needs to be proportional.

So, 10 : 1450 :: 108 : x

First, find the number of bricks needed for 1 foot of wall:

$\dfrac{1450}{10} = 145$ bricks per foot

Now, multiply by the total length of the walls:

145 × 108 = 15660

Hence, the mason would need approximately 15660 bricks to build the house.

Puneeth's father went from Lucknow to Kanpur in 2 hours by riding his motorcycle at a speed of 50 km/h. If he drives at 75 km/h, how long will it take him to reach Kanpur? Can we form this problem as a proportion—

50 : 2 :: 75 : __

Would it take Puneeth's father more time or less time to reach Kanpur? Think about it.

Answer

Given:

Speed = 50 km/h

Time taken = 2 hours

Distance travelled = Speed × Time

= 50 × 2

= 100 km

If the speed is 75 km/h, then

Time = $\dfrac{\text{Distance}}{\text{Speed}}$

$= \dfrac{100}{75} \\[1em] = \dfrac{4}{3}\text{ hours} \\[1em] = 1\dfrac{1}{3}\text{ hours} \\[1em] = 1\text{ hour 20 minutes}.$

Now consider the proportion the proportion 50 : 2 :: 75 : __

As the speed increases from 50 km/h to 75 km/h, the time taken should decrease.

But in a proportion, when one quantity increases, the other quantity also increases by the same factor. Here, one quantity increases while the other decreases.

Therefore, speed and time are not directly proportional.

So, this situation cannot be represented by the proportion 50 : 2 :: 75 : __

Hence, Puneeth's father will take $\bold {1\dfrac{1}{3}}$ hours (1 hour 20 minutes) to reach Kanpur, and the proportion 50 : 2 :: 75 : __ cannot be used because speed and time are inversely proportional.

Observe the table given below.

The ratio of the volume of a sachet to a small bottle is 6 : 180, and the ratio of their prices is 2 : 154. Are these ratios proportional? Why do you think the ratio of the prices is not proportional to the ratio of the volumes?

Answer

Let us check whether the two ratios are proportional by reducing them to their simplest form.

Ratio of volumes (sachet : small bottle) = 6 : 180.

Dividing both terms by their HCF 6:

$6 : 180 = \dfrac{6 \div 6}{180 \div 6} = \dfrac{1}{30} = 1 : 30$

Ratio of prices (sachet : small bottle) = 2 : 154.

Dividing both terms by their HCF 2:

$2 : 154 = \dfrac{2 \div 2}{154 \div 2} = \dfrac{1}{77} = 1 : 77$

Comparing the simplest forms, 1 : 30 ≠ 1 : 77

Therefore, the ratios are not proportional.

The price of a shampoo container depends not only on the amount of shampoo but also on the cost of the packet or bottle, manufacturing, transport and other expenses. Larger containers are also often sold at a lower cost per millilitre.

Hence, the ratios 6 : 180 and 2 : 154 are not proportional, because price depends on factors other than volume of shampoo.

If you want to share 12 counters between two people in the ratio of 3 : 1, how many counters would each of them get?

Answer

Given:

Total counters = 12

Ratio = 3 : 1

Find total parts: 3 + 1 = 4 parts.

Find the value of 1 part: $\dfrac{12}{4}$ = 3 counters.

Now calculate each share by multiplying with 3 counters:

First person gets: 3 × 3 counters = 9 counters

Second person gets: 1 × 3 counters = 3 counters

Hence, one person gets 9 counters and the other gets 3 counters.

If you want to share 42 counters between two people in the ratio of 4 : 3, how will you do it?

Answer

Given:

Total counters = 42

Ratio = 4 : 3

Split the counters into groups so that the first person gets 4 groups and the second person gets 3 groups.

Find total groups: 4 + 3 = 7 groups.

Find the size of each group: $\dfrac{42}{7}$ = 6 counters.

Now calculate each share by multiplying with 6 counters:

First person gets: 4 × 6 counters = 24 counters

Second person gets: 3 × 6 counters = 18 counters

Hence, one person gets 24 counters and the other gets 18 counters.

Prashanti and Bhuvan started a food cart business near their school. Prashanti invested ₹75,000 and Bhuvan invested ₹25,000. At the end of the first month, they gained a profit of ₹4,000. They decided that they would share the profit in the same ratio as that of their investment. What is each person's share of the profit?

Answer

Given:

Prashanti's investment = ₹75,000

Bhuvan's investment = ₹25,000

Total profit = ₹4,000

The profit is shared in the ratio of their investments, i.e., 75000 : 25000.

Reduce this ratio to its simplest form by dividing both terms by their HCF 25000:

$75000 : 25000 = \dfrac{75000 \div 25000}{25000 \div 25000} = \dfrac{3}{1} = 3 : 1$

Find total parts: 3 + 1 = 4 parts.

Find the value of 1 part: ₹ $\dfrac{4000}{4}$ = ₹1000.

Now calculate each share by multiplying with ₹1000:

Prashanti's share: 3 × ₹1000 = ₹3000

Bhuvan's share: 1 × ₹1000 = ₹1000

Hence, Prashanti gets ₹3000 and Bhuvan gets ₹1000 of the profit.

Divide ₹4,500 into two parts in the ratio 2 : 3.

Answer

Given:

Total money = ₹4500

Ratio = 2 : 3

Find total parts: 2 + 3 = 5 parts.

Find the value of 1 part: ₹ $\dfrac{4500}{5}$ = ₹900.

Now calculate each part by multiplying with ₹900:

First part = 2 × ₹900 = ₹1800

Second part = 3 × ₹900 = ₹2700

Hence, the two parts are ₹1800 and ₹2700.

In a science lab, acid and water are mixed in the ratio of 1 : 5 to make a solution. In a bottle that has 240 mL of the solution, how much acid and water does the solution contain?

Answer

Given:

Total solution = 240 mL

Ratio of acid to water = 1 : 5

Find total parts: 1 + 5 = 6 parts.

Find the value of 1 part: $\dfrac{240 \text{ mL}}{6}$ = 40 mL.

Now calculate each quantity by multiplying with 40 mL:

Acid = 1 × 40 mL = 40 mL

Water = 5 × 40 mL = 200 mL

Hence, the solution contains 40 mL of acid and 200 mL of water.

Blue and yellow paints are mixed in the ratio of 3 : 5 to produce green paint. To produce 40 mL of green paint, how much of these two colours are needed? To make the paint a lighter shade of green, I added 20 mL of yellow to the mixture. What is the new ratio of blue and yellow in the paint?

Answer

Given:

Total green paint = 40 mL

Ratio of blue to yellow = 3 : 5

Find total parts: 3 + 5 = 8 parts.

Find the value of 1 part: $\dfrac{40 \text{ mL}}{8}$ = 5 mL.

Now calculate each quantity by multiplying with 5 mL:

Blue paint = 3 × 5 mL = 15 mL

Yellow paint = 5 × 5 mL = 25 mL

Next, 20 mL of yellow is added to make a lighter shade.

New quantity of yellow = 25 mL + 20 mL = 45 mL.

Blue paint remains the same = 15 mL.

New ratio of blue to yellow = 15 : 45.

Reduce to the simplest form by dividing both terms by their HCF 15:

$15 : 45 = \dfrac{15 \div 15}{45 \div 15} = \dfrac{1}{3} = 1 : 3$

Hence, 15 mL of blue and 25 mL of yellow are needed, and the new ratio of blue to yellow is 1 : 3.

To make soft idlis, you need to mix rice and urad dal in the ratio of 2 : 1. If you need 6 cups of this mixture to make idlis tomorrow morning, how many cups of rice and urad dal will you need?

Answer

Given:

Total mixture = 6 cups

Ratio of rice to urad dal = 2 : 1

Find total parts: 2 + 1 = 3 parts.

Find the value of 1 part: $\dfrac{6}{3}$ = 2 cups.

Now calculate each quantity by multiplying with 2 cups:

Rice = 2 × 2 cups = 4 cups

Urad dal = 1 × 2 cups = 2 cups

Hence, 4 cups of rice and 2 cups of urad dal will be needed.

I have one bucket of orange paint that I made by mixing red and yellow paints in the ratio of 3 : 5. I added another bucket of yellow paint to this mixture. What is the ratio of red paint to yellow paint in the new mixture?

Answer

Given:

Ratio of red to yellow in the orange paint = 3 : 5

Total parts in one bucket: 3 + 5 = 8 parts.

So, in one bucket of orange paint:

Part of red paint = $\dfrac{3}{8}$ of a bucket.

Part of yellow paint = $\dfrac{5}{8}$ of a bucket.

Now, one full bucket of yellow paint is added.

New quantity of yellow paint = $\dfrac{5}{8} + 1 = \dfrac{5 + 8}{8} = \dfrac{13}{8}$ of a bucket.

The red paint stays the same = $\dfrac{3}{8}$ of a bucket.

New ratio of red to yellow = $\dfrac{3}{8} : \dfrac{13}{8}$

Multiply both terms by 8 to remove the fractions:

$\dfrac{3}{8} : \dfrac{13}{8} = 3 : 13$

Hence, the new ratio of red paint to yellow paint is 3 : 13.

Anagh mixes 600 mL of orange juice with 900 mL of apple juice to make a fruit drink. Write the ratio of orange juice to apple juice in its simplest form.

Answer

Given:

Orange juice = 600 mL

Apple juice = 900 mL

Ratio of orange juice to apple juice = 600 : 900

Let us find the HCF of 600 and 900:

$\begin{array}{r} 1 \\ 600 \overline{) 900} \\ \underline{-600} \\ 300 \overline{) 600} ( 2 \\ \underline{-600} \phantom{1)} \\ 0 \phantom{1)} \end{array}$

HCF = 300

∴ 600 : 900 = $\dfrac{600 \div 300}{900 \div 300} = \dfrac{2}{3} = 2 : 3$

Hence, the ratio of orange juice to apple juice is 2 : 3.

Last year, we hired 3 buses for the school trip. We had a total of 162 students and teachers who went on that trip and all the buses were full. This year we have 204 students. How many buses will we need? Will all the buses be full?

Answer

Given:

3 buses carried 162 students and teachers, and all the buses were full.

Capacity of one bus = 162 ÷ 3 = 54 persons.

Let x be the number of buses needed for 204 students and teachers.

3 : 162 :: x : 204

Buses needed: x = $\dfrac{204 \times 3}{162} = \dfrac{204}{54} = 3\dfrac{42}{54} ≈ 3.78$

Since a part of a bus still requires one full bus, we need 4 buses.

Number of seats in 4 buses = 4 × 54 = 216

Total number of persons = 204

Vacant seats = 216 – 204 = 12

Hence, 4 buses will be needed and the buses will not all be full; 12 seats will remain vacant.

The area of Delhi is 1,484 sq. km and the area of Mumbai is 550 sq. km. The population of Delhi is approximately 30 million and that of Mumbai is 20 million people. Which city is more crowded? Why do you say so?

Answer

To compare how crowded the cities are, find the number of persons in 1 sq. km of each city.

Number of persons per sq. km in Delhi = $\dfrac{30,000,000}{1,484} \approx 20,216$

Number of persons per sq. km in Mumbai = $\dfrac{20,000,000}{550} \approx 36,364$

Since the number of persons per sq. km in Mumbai is more than in Delhi, Mumbai is more crowded.

Hence, Mumbai is more crowded, because it has more persons per square kilometre.

A crane of height 155 cm has its neck and the rest of its body in the ratio 4 : 6. For your height, if your neck and the rest of the body also had this ratio, how tall would your neck be?

Answer

Given:

Ratio of neck to the rest of the body = 4 : 6

Total parts = 4 + 6 = 10 parts.

So the neck is 4 parts out of 10 parts of the total height.

∴ Neck height = $\dfrac{4}{10} \times (\text{your height})$

Hence, your neck would be $\bold {\dfrac{4}{10}}$ of your height.

Let us try an ancient problem from Lilavati. At that time weights were measured in a unit named palas and niskas was a unit of money. "If $2\dfrac{1}{2}$ palas of saffron costs $\dfrac{3}{7}$ niskas, O expert businessman! tell me quickly what quantity of saffron can be bought for 9 niskas?"

Answer

Given:

$2\dfrac{1}{2}$ palas, i.e. $\dfrac{5}{2}$ palas of saffron costs $\dfrac{3}{7}$ niskas.

Cost of 1 pala = $\dfrac{3}{7} \div \dfrac{5}{2} = \dfrac{3}{7} \times \dfrac{2}{5} = \dfrac{6}{35}$ niskas.

Quantity of saffron for 9 niskas = $9 \div \dfrac{6}{35}$

$⇒ 9 \times \dfrac{35}{6} = \dfrac{315}{6} = 52.5$ palas

Hence, 52.5 palas of saffron can be bought for 9 niskas.

Harmain is a 1-year-old girl. Her elder brother is 5 years old. What will be Harmain's age when the ratio of her age to her brother's age is 1 : 2?

Answer

Given:

Harmain's present age = 1 year and her brother's present age = 5 years.

Let after x years the ratio of Harmain's age to her brother's age be 1 : 2.

x + 1 : x + 5 = 1 : 2

$\dfrac{x + 1}{x + 5} = \dfrac{1}{2}$

⇒ 2(x + 1) = x + 5

⇒ 2x + 2 = x + 5

⇒ 2x - x = 5 - 2

⇒ x = 3

So this happens after 3 years.

Harmain's age = 1 + 3 = 4 years.

Hence, Harmain will be 4 years old.

The mass of equal volumes of gold and water are in the ratio 37 : 2. If 1 litre of water is 1 kg in mass, what is the mass of 1 litre of gold?

Answer

Given:

Mass of 1 litre of water = 1 kg.

Ratio of mass of equal volumes of gold to water = 37 : 2.

Let the mass of 1 litre of gold = x kg.

37 : 2 :: x : 1

⇒ x = $\dfrac{37 \times 1}{2} = \dfrac{37}{2} = 18.5$ kg

Hence, the mass of 1 litre of gold is 18.5 kg.

It is good farming practice to apply 10 tonnes of cow manure for 1 acre of land. A farmer is planning to grow tomatoes in a plot of size 200 ft by 500 ft. How much manure should he buy? (Please refer to the section on Unit Conversions earlier in this chapter.)

Answer

Given:

Cow manure required for 1 acre = 10 tonnes

We know that 1 tonne = 1000 kg, so 10 tonnes = 10,000 kg.

Also, 1 acre = 43,560 sq. ft.

So, the ratio of cow manure to area of land is 10000 : 43560.

Size of plot = 200 × 500 = 1,00,000 sq. ft.

Let the cow manure required be x kg.

Then, x : 100000 :: 10000 : 43560

⇒ x = $\dfrac{100000 \times 10000}{43560} ≈ 22956.8$ kg

Hence, the farmer should buy about 22956.8 kg (≈ 22.9568 tonnes) of cow manure.

A tap takes 15 seconds to fill a mug of water. The volume of the mug is 500 mL. How much time does the same tap take to fill a bucket of water if the bucket has a 10-litre capacity?

Answer

Given:

Time to fill one mug of 500 mL = 15 seconds.

Capacity of bucket = 10 litres.

We know that 1 L = 1,000 mL, so 10 L = 10,000 mL.

Number of mugs needed to fill the bucket = $\dfrac{10,000}{500} = 20$ mugs.

Time to fill the bucket = 20 × 15 = 300 seconds = 5 minutes.

Hence, the tap takes 300 seconds, i.e. 5 minutes, to fill the bucket.

One acre of land costs ₹15,00,000. What is the cost of 2,400 square feet of the same land?

Answer

Given:

1 acre = 43,560 sq. ft.

Cost of 43,560 sq. ft. of land = ₹15,00,000

Let the cost of 2,400 sq. ft. of land be ₹x.

Since the area and cost are directly proportional,

43560 : 1500000 :: 2400 : x

Cost of 2,400 sq. ft = $\dfrac{1500000 \times 2400}{43560} ≈ ₹82645$

Hence, the cost of 2,400 square feet of land is approximately ₹82,645.

A tractor can plough the same area of a field 4 times faster than a pair of oxen. A farmer wants to plough his 20-acre field. A pair of oxen takes 6 hours to plough an acre of land. How much time would it take if the farmer used a pair of oxen to plough the field? How much time would it take him if he decides to use a tractor instead?

Answer

Given:

The tractor is 4 times faster than a pair of oxen, so the ratio of time taken by tractor to oxen = 1 : 4.

Time taken by a pair of oxen to plough 1 acre = 6 hours.

Time taken by a tractor to plough 1 acre = $\dfrac{6}{4} = 1.5$ hours.

Time taken by a pair of oxen to plough the 20-acre field = 20 × 6 = 120 hours.

Time taken by the tractor to plough the 20-acre field = 20 × 1.5 = 30 hours.

Hence, the pair of oxen would take 120 hours, while the tractor would take 30 hours.

The ₹10 coin is an alloy of copper and nickel called 'cupro-nickel'. Copper and nickel are mixed in a 3 : 1 ratio to get this alloy. The mass of the coin is 7.74 grams. If the cost of copper is ₹906 per kg and the cost of nickel is ₹1,341 per kg, what is the cost of these metals in a ₹10 coin?

Answer

Given:

Mass of the coin = 7.74 grams.

Ratio of copper to nickel = 3 : 1.

Total parts = 3 + 1 = 4 parts.

Value of 1 part = $\dfrac{7.74}{4} = 1.935$ g.

Mass of copper = 3 × 1.935 = 5.805 g

Mass of nickel = 1 × 1.935 = 1.935 g

Cost of copper (₹906 per kg, i.e. per 1,000 g):

Cost of 5.805 g of copper = $\dfrac{906 \times 5.805}{1000} ≈ ₹5.26$

Cost of nickel (₹1,341 per kg, i.e. per 1,000 g):

Cost of 1.935 g of nickel = $\dfrac{1341 \times 1.935}{1000} ≈ ₹2.59$

Total cost of metals = ₹5.26 + ₹2.59 = ₹7.85

Hence, the copper costs about ₹5.26 and the nickel costs about ₹2.59, making the cost of the metals in a ₹10 coin about ₹7.85.