PROPORTIONAL REASONING-1
KEY POINTS & FIGURE IT OUT
7.2 Ratios
In a ratio of the form a : b, we can say that for every 'a' units of the first quantity, there are 'b' units of the second quantity.
A more systematic way to compare whether the ratios are proportional is to reduce them to their simplest form and see if these simplest forms are the same.
7.3 Ratios in their Simplest Form
We can reduce ratios to their simplest form by dividing the terms by their HCF.
When two ratios are the same in their simplest forms, we say the ratios are proportional.
We use the symbol :: to show proportionality.
Example:
60 : 40 :: 30 : 20 and 60 : 40 :: 90 : 60.
7.4 Problem Solving with Proportional Reasoning
Example 1:
Are the ratios 3 : 4 and 72 : 96 proportional?
3 : 4 is already in its simplest form. To find the simplest form of 72 : 96, we need to divide both terms by their HCF.
The HCF of 72 and 96 is 24. Dividing both terms by 24, we get 3 : 4. Since both ratios in their simplest form are the same, they are proportional
Example 2:
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? 6 : 10 :: 18 : ?
The ratio of glasses of lemonade to spoons of sugar is 6 : 10.
6 : 10 :: 18 : x.
6 : 10 :: 18 : x
\[
\frac{6}{10} = \frac{18}{x} \implies x = \frac{18 \times 10}{6} = 30
\]
So, she should use 30 spoons of sugar to make 18 glasses of lemonade with the same sweetness as earlier
Example 3:
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?
The ratio in Nitin’s case is 60 : 3, That is , 20 : 1 (in its simplest form).
The ratio in Hari’s case is 40 : 2, That is , 20 : 1 (in its simplest form).
Since both ratios are proportional, the walls are equally strong. Nitin should not worry!
Example 4:
In my school, there are 5 teachers and 170 students. The ratio of teachers to students in my school is 5 : 170. Count the number of teachers and students in your school. What is the ratio of teachers to students in your school? Write it below. ______ : ______ Is the teacher-to-student ratio in your school proportional to the one in my school?
Example:
My school = 5:170
Your school = 6: 240
5:170::6:240 no not in proportional
Example 5:
Measure the width and height (to the nearest cm) of the blackboard in your classroom. What is the ratio of width to height of the blackboard? ______ : ______ Can you draw a rectangle in your notebook whose width and height are proportional to the ratio of the blackboard? Compare the rectangle you have drawn to those drawn by your classmates. Do they all look the same?
Black board Width: 120 cm Height: 90 cm HCF = 30
120:90 = 4:3
Draw rectangle with Width = 8 cm
Height = (8÷4)×3=6 cm
Yes, in terms of shape. All rectangles drawn with a 4 : 3 ratio will have the same shape. They will look like scaled versions of each other—some may be larger, some smaller, but the proportion of width to height will be identical. This is the essence of similarity. they look different in size, Precision, drawing accuracy.
Example 6:
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?
The ratio of Neelima’s age to her mother’s age when Neelima is 3 years old is 3 : 30 (her mother’s age is 10 times Neelima’s age).
In the simplest form, it is 1 : 10.
When Neelima is 12 years old (, 9 years later), the ratio of their ages will be 12 : 39 (9 years later, her mother would be 39 years old).
In the simplest form, it is 4 : 13.
When we add (or subtract) the same number from the terms of a ratio, the ratio changes and is not necessarily proportional to the original ratio.
Example 7:
Fill in the missing numbers for the following ratios that are proportional to
14 : 21.
______ : 42
6 : ______
2 : ______What factor should we multiply 14 by to get 6? Can it be an integer? Or should it be a fraction?
14:21::28:42
14y = 6
Factor y = \( \frac{6}{14} = \frac{3}{7} \)
→ \( 21 \times \frac{3}{7} = 9 \)
Ratio = 6 : 9.
2. ? : 42
Factor = \( \frac{42}{21} = 2 \)
→ \( 14 \times 2 = 28 \)
So, the ratio is 6 : 9.
Ratio = 28 : 42
In the third ratio, the first term is 2.
14 : 21 divide by 7 (HCF of 14 and 21) → 2 : 3.
So, the ratio is 2 : 3
7.5 Filter Coffee Example
Filter Coffee! Filter coffee is a beverage 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. In this case, we can say that the ratio of coffee decoction to milk is 15 : 35. 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? And when they want ‘lighter’ filter coffee, he mixes 10 ml of coffee and 40 ml of milk, making the ratio 10 : 40.Why is this coffee lighter? The following table shows the different ratios in which Manjunath mixes coffee decoction with milk. Write in the last column if the coffee is stronger or lighter than the regular coffee
Why is the 20:30 coffee stronger?
Because the coffee-to-milk ratio 2:32:3 (≈0.667≈0.667) is larger than 3:73:7 (≈0.429≈0.429), meaning more coffee per unit of milk.
Why is the 10:40 coffee lighter?
Because the coffee-to-milk ratio 1:4 (=0.25) is smaller than 3:7 (≈0.429), meaning less coffee per unit of milk.
Regular coffee: 15 ml decoction + 35 ml milk → ratio 15 : 35 → simplest 3 : 7.
Stronger: 20 ml decoction + 30 ml milk → ratio 20 : 30 → simplest 2 : 3.
Lighter: 10 ml decoction + 40 ml milk → ratio 10 : 40 → simplest 1 : 4.
Comparison:
-
2 : 3 (≈0.667) > 3 : 7 (≈0.429) → stronger.
1 : 4 (0.25) < 3 : 7 (≈0.429) → lighter.
|
Coffee Decoction (in mL) |
Milk (in mL) |
Regular/Strong/Light |
|
300 |
600 |
Stronger |
|
150 |
500 |
Lighter |
|
200 |
400 |
Stronger |
|
24 |
56 |
Regular |
|
100 |
300 |
Lighter |
Figure it Out (Page 165-167)
1. 1. Circle the following statements of proportion that are true. (i) 4 : 7 :: 12 : 21 (iii) 7 : 12 :: 12 : 7 (v) 12 : 18 :: 28 : 12 (ii) 8 : 3 :: 24 : 6 (iv) 21 : 6 :: 35 : 10 (vi) 24 : 8 :: 9 : 3
Solution:(1) Given statement is 4:7:: 12:21.
This is true if 4/7 = 12/21 which is true. or if 4/7 = 4/7 ... Given statement is true,
(ii) Given statement is 8:3:: 24: 6.
This is true if 8/3 = 24/6 which is false. or if 8/3 = 4 ... Given statement is not true.
(iii) Given statement is 7:12::12:7.
This is true if which is false. Given statement is not true. 7/12 = 12/7
(iv) Given statement is 21:6:: 35:10.
This is true if or if 7/2 = 7/2 which is true. .. Given statement is true.
(v) Given statement is 12:18:28:12.
This is true if 12/18 = 28/12 or if 2/3 = 7/3 or 27, which is false.
Given statement is not true,
(vi) Given statement is 24:8::9:3.
This is true if 24/8 = 9/3 or if 33, which is true.... Both simplify to 3:1 Given statement is 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 ✅
2. 2. Give 3 ratios that are proportional to 4 : 9. ______ : ______ ______ : ______ ______ : ______
Multiply both terms by same factor: e.g., 8:18, 12:27, 20:45
4/9 = (4 * 2)/(9 * 2) = (4 * 3)/(9 * 3) = (4 * 4)/(9 * 4)
4/9 = 8/18 = 12/27 = 16/36
4:9::8:18, 4:9::12:27 and 4:9::16:36
8 : 18, 12 : 27, 20 : 45
3. Fill in the missing numbers for these ratios that are proportional to 18 : 24. 3 : ______ 12 : ______ 20 : ______ 27 : ______
(1) Given ratio is 18: 24, let 18:24::3:x
18/24 = 3/x
3/4 = 3/x
x = 4
let 18:24:12: x.
18/24 = 12/x or
3x = 48 or or
x = 48/3 = 16
3/4 = 12/x
Missing number in the ratio 12: is 16.
(iii) let 18:24:20: x.
or 3x = 80
x = 80/3
3/4 = 20/x
18/24 = 20/x
Missing number in the ratio 20: _is 80/3 *
(iv) let 18:24:27:x. or 3x = 108 or Missing number in the ratio 27: is 36
x = 108/3 = 36
3/4 = 27/x *
18/24 = 27/x
3 : 4, 12 : 16, 20 : \( \frac{80}{3} \), 27 : 36
4. 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.
Solution:
Using a scale, we measure the width and height of given rectangles.
The ratio 'Width: Height' for given rectangles A, B, C, D and E are respectively 1:3, 3:2, 9:4, 7:2 and 3:1. These ratios are all distinct. The ratios of A and E are 1:3 and 3: 1 respectively
Rectangles with same width : height ratio are similar. 
5. 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?
Width 32 mm and height 18 mm
Ratio is 32:18.
New width = 1/2 X 32 =16mm
new height = 1/2 x18 = 9 mm
New similar rectangle is shown in the figure.
let 'factor of change' be 2.
New width = 2x32= 64 mm and
new height = 2 × 18 = 36 mm
Factor of change = 2:
Smaller: 16 mm × 9 mm
Larger: 64 mm × 36 mm
6. 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: Ratio of grey bricks to coloured bricks in pattern:
Number of grey bricks in one set of pattern = 2 + 3 + 4 = 9
Number of coloured bricks in one set of pattern = 3 + 2 + 1 = 6
Ratio of grey bricks to coloured bricks 9:6 = 3:2
Ratio in the simplest form 3:2
(b) One set of pattern
Number of grey bricks in one set of pattern
= (½ + 1 + 1 + ½) + (1 + 1) + (½ + 1 + ½) + (1 + 1) + (½ + 1 + ½) + (1 + 1) + (½ + 1 + 1 + ½)
= 3 + 2 + 2 + 2 + 2 + 2 + 3 = 16
Number of coloured bricks in one set of pattern = 1 + (1 + 1) + (1 + 1) + (1 + 1) + (1 + 1) +(1+1)+1 Ratio of grey bricks to coloured bricks = 16:12 = 4:3 .
Ratio in the simplest form = 4:3 .
(a) Grey : Coloured = 9 : 6 → simplest = 3 : 2
(b) Grey : Coloured = 16 : 12 → simplest = 4 : 3
7. let us draw some human figures. Measure your friend’s body — the lengths of their head, torso, arms, and legs. Write the ratios as mentioned below— head : torso ______ : ______ torso : arms ______ : ______ torso : legs ______ : ______Now, draw a figure with head, torso, arms, and legs with equivalent ratios as above. Does the drawing look more realistic if the ratios are proportional? Why? Why not? Human figure proportions:
head : torso 25:60 (Answer: 5 : 12)
torso : arms 60:65 (Answer: 12 : 13)
torso : legs 60:80 (Answer: 3 : 4)
Now I draw: Head: 5 cm tall, Torso: 12 cm tall, Arms: 13 cm each, Legs: 16 cm each, This will be a proportional stick figure or simplified human shape.
Does it look more realistic if the ratios are proportional?
Yes, generally it does look more realistic.
Why? Human bodies have consistent proportional relationships across individuals (though they vary somewhat). If you keep the torso-to-leg ratio realistic (e.g., around 3:4 or similar), and head-to-torso ratio correct (around 1:3 to 1:4), the figure will look naturally human.
If you make the head too big compared to the torso (e.g., 1:1 ratio), it will look cartoonish or child-like. If legs are too short compared to the torso, it looks unnatural. Proportionality based on real human measurements captures the natural balance of body parts, making the drawing lifelike.
Why not (possible counterpoint)?
Sometimes in art, especially cartoons or stylized figures, artists intentionally break proportions for expressive effect (big head for cuteness, long legs for elegance). So, “realistic” isn’t always the goal in art—but if the goal is a realistic human figure, keeping ratios proportional is key.
Final thought for the student:
When you draw using the actual ratios from your friend, you are drawing a figure that matches their specific body proportions. This will look like a recognizable representation of that person, which is what “realistic” means in this activity.
Example: head : torso = 25 : 60 → 5 : 12
Proportional drawing looks realistic because body parts maintain natural ratios.
7.6 Rule of Three (Trairasika)
Two ratios are proportional if their terms are equal when cross multiplied. The fourth unknown quantity can be found through such cross multiplication. a : b :: c : d.
In ancient India, ΔryabhaαΉa (199 CE) and others called such problems of proportionality Rule of Three problems.
There were 3 numbers given — the pramΔαΉa (measure — ‘a’ in our case), the phala (fruit — ‘b’ in our case), and the ichchhΔ (requisition — ‘c’ in our case). To find the ichchhΔphala (yield — ‘d’ in our case),
ΔryabhaαΉa says, “Multiply the phala by the ichchhΔ and divide the resulting product by the pramΔαΉa.”
In other words, ΔryabhaαΉa says, “pramΔαΉa : phala :: ichchhΔ : ichchhΔphala,”
therefore, pramΔαΉa × ichchhΔphala = phala × ichchhΔ.
ichchhΔphala = "phala × ichchhΔ" /"pramΔαΉa" . Using the cross multiplication method proposed by ΔryabhaαΉa, ancient Indians solved complex problems that involved proportionality
If a : b :: c : d, then
\[
d = \frac{b \times c}{a}
\]
Ancient Indian method:
- pramΔαΉa (a)
- phala (b)
- ichchhΔ (c)
- ichchhΔphala (d) = \( \frac{b \times c}{a} \)
Example 8:
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? The ratio of the number of students to the amount of rice needs to be proportional. So, 120 : 15 :: 80 : ? What is the factor of change in the first term?
For 120 students, required rice is 15 kg
Let x kg of rice be required for 80 students
Ratios 120:15 and 80 : x are in proportion
120 : 15 : : 80 : x
by dividing the terms 80 :120 = 2 : 3.
The number of students is reduced by a factor of 2 : 3 .
On multiplying the weight of rice by the same factor, we get, 15 × 2 /3 = 10.
School: 120 students → 15 kg rice.
80 students → ? kg rice.
120 : 15 :: 80 : x
\[
x = \frac{15 \times 80}{120} = 10 \ \text{kg}
\]
So, the cook should make 10 kg of rice on that day
Example 9:
(i) A car travels 90 km in 150 minutes. If it continues at the same speed, what distance will it cover in 4 hours? If it continues at the same speed, the ratio of the time taken should be proportional to the ratio of the distance covered.
(ii) 150 : 90 :: 4 : ? Is this the right way to formulate the question?
(iii) How can you find the distance covered in 240 minutes?
(i) 4 hours = 4 x 60 = 240 minutes
In 150 minutes, distance covered 90 km
Let x km be covered in 4 hours in 240 minutes
The ratios 150:90 and 240:x are in proportion.
150:90::240:x
(ii) unit must be same in comparing ratios
so 150:90::4:x is meaningless.
so 150:90::4:x is meaningless.
and150:90::240:x is correct.
(iii) 150 : 90 :: 240 : x.
By cross multiplication, we get 150 × x = 240 × 90
Therefore, x = 144.
Car: 90 km in 150 min. Distance in 4 hours (240 min)?
150 : 90 :: 240 : x
\[
x = \frac{90 \times 240}{150} = 144 \ \text{km}
\]
The distance covered by the car in 4 hours is 144 km
Example 10:
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? Which tea is more expensive? Why?
The ratio of weight to price of the Himachal tea is 200 : 200.
So, the weight to price ratio is 1000 : 800 in Meghalaya after we convert the weight to grams.
The Himachal tea ratio in its simplest form is 1 : 1.
The Meghalaya tea ratio in its simplest form is 5 : 4.
So, the ratios are not proportional.
The price of 1 kg of tea is x rupees.
200 g is 1/5 of 1 kg. So, 1/5 x = 200.
1/5 x × 5 = 200 × 5
x = 1000. So, the cost of 1 kg of tea is ₹800 in Meghalaya and ₹1,000 in Himachal Pradesh. Therefore, the tea from Himachal Pradesh is more expensive.
Himachal tea: 200 g → ₹200 → ratio 1 : 1.
Meghalaya tea: 1 kg (1000 g) → ₹800 → ratio 5 : 4.
Not proportional.
Cost per kg:
Himachal = ₹1000, Meghalaya = ₹800 → Himachal more expensive.
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###
Figure it Out (page 170 - 171)
1. The Earth travels approximately 940 million kilometres around the Sun in a year. How many kilometres will it travel in a week?
Solution:
1 million = 10 lakh = 10,00,000
1 year = \( \frac{365}{7} \) weeks
940 million kilometres = 940 X 10 ,00,000 kilometres are travelled by the Earth in 1 year.
let x kilometres be travelled by the Earth in 1 week
The ratios 940 × 10,00,000: \( \frac{365}{7} \) and x: 1 are in proportion.
\[ x = \frac{940 \times 1000000}{365/7} \] = $ \frac{x}{1} $
\[ x = \frac{188 \times 7000000}{73} \]
In 1 week, Earth travels nearly 1,80,27,397 kilometres around the Sun.
2. 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 Proportional Reasoning-1 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.
Total length of walls =
= AI + CH + DE + FG + IG + AF + CD
=12 + (9 + 12) + 9 + 12 + (9 + 15) + (9 + 15) + 6 = 108ft
let x bricks be required for 108 ft long wall.
Ratio of length of wall to number of bricks = 108 : x
These ratios are in proportion. 10:1450::108: x
\( \frac{10}{1450} \) = \( \frac{108}{x} \)
X = 145 x 108 = 15660
Number of required bricks = 15660
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. Even though this problem looks similar to the previous problems, it cannot be solved using the Rule of Three! The time of travel would actually decrease when the speed increases. So this problem cannot be modelled as 50 : 2 :: 75 : __
Solution:
Time taken at the speed of 50 km/h = 2 hours ..
Distance = Speed × Time = 50 × 2 = 100 km
At speed of 75km / h time taken = \( \frac{100}{75} \) = $ \frac{4}{3} $ hours
The ratios 50: 2 and proportion. 75 : \( \frac{4}{3} \) are not in proportion.
We cannot write: 50:2 :: 75 : $ \frac{4}{3} $
At speed of 75km / h Puneeth's father will take $ \frac{4}{3} $ hours to reach Kanpur, which is less than 2 hours.
if we want to divide a quantity x in the ratio of m: n,
then the parts will be 
Example 11:
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?
The ratio of their investment is 75000 : 25000.
Reducing this ratio to its simplest form, we get 3 : 1.
3 + 1 is 4 and dividing the profit of 4000 by 4, we get 1000.
So, Prashanti’s share is 3 × 1000 and Bhuvan’s share is 1 × 1000.
So, Prashanti would get ₹3,000 and Bhuvan would get ₹1,000 of the profit.
Example 12:
A mixture of 40 kg contains sand and cement in the ratio of 3 : 1. How much cement should be added to the mixture to make the ratio of sand to cement 5 : 2?
let us find the quantity of sand and cement in the original mixture.
The ratio is 3 : 1 and the total weight is 40 kg.
So, the weight of sand is 3/4 x 40 = 30 kg.
The weight of cement is 1/4 x 40 = 10 kg.
The weight of sand is the same in the new mixture. It remains 30.
But the new ratio of sand to cement is 5 : 2.
So the question is, 5 : 2 :: 30 : ?
If the ratio is 5 : 2, then the second term is 2 5 times the first term.
Since the new ratio is equivalent to 5 : 2, the second term in the new ratio should also be 2 /5 times of 30.
2/5 × 30 = 12.
The new mixture should have 12 kg of cement if the ratio of sand to cement is to be 5 : 2.
There is 10 kg of cement already. So, we need to add 2 kg of cement to the original mixture
Mixture: 40 kg, sand : cement = 3 : 1 → sand = 30 kg, cement = 10 kg.
New ratio = 5 : 2, sand unchanged (30 kg).
Cement needed = \( \frac{2}{5} \times 30 = 12 \) kg → add 2 kg.
Figure it Out (Page 175)
1. Divide ₹4,500 into two parts in the ratio 2 : 3
Given ratio = 2 : 3
Amount to be divided = ₹4,500 .
First part = 2/5 × 4,500 = 2 x 900 = ₹1,800
Second part = 3/5 × 4,500 = 3 x 900 = ₹2,700
Two parts are ₹1,800 and ₹2,700,
2. 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?
Solution:
Ratio of acid and water = 1 : 5
Quantity of solution = 240 ml.
Quantity of acid = 1/6 x 240 = 40 ml
Quantity of water = 5/6 x 240 = 200 ml
Quantity of acid and water in the solution are 40 ml and 200 ml.
Acid = \( \frac{1}{6} \times 240 = 40 \) ml
Water = \( \frac{5}{6} \times 240 = 200 \) ml.
3. 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?
Solution:
Ratio of blue and yellow paints = 3 : 5
Quantity of green paint = 40 ml.
Quantity of blue paint = 3/8 × 40 = 15mI
Quantity of yellow paint = 5/8 x 40 = 25ml
Addition of yellow paint to the mixture = 20 ml
New quantity of blue paint = 15 ml
New quantity of yellow paint = 25ml + 20ml = 45ml
New ratio of blue and yellow paints =15 : 45 = 1 :3
Blue = \( \frac{3}{8} \times 40 = 15 \) ml
Yellow = \( \frac{5}{8} \times 40 = 25 \) ml.
Add 20 ml yellow → new ratio = 15 : 45 = 1 : 3.
4. 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?
Solution:
Ratio of rice and urad dal 2 :1
Total number of cups of mixture = 6
Number of cups of rice = 2/3 x 6 = 4
Number of cups of urad dal = 1/3 x 6 = 2
4 cups of rice and 2 cups of urad dal are to be mixed.
Rice = \( \frac{2}{3} \times 6 = 4 \) cups
Urad dal = \( \frac{1}{3} \times 6 = 2 \) cups.
5. 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?
Solution:
let capacity of one bucket be x litre.
Ratio of red paint and yellow paint = 3:5
Quantity of red paint in the bucket = 3/8 x π₯ = 3π₯/8
Quantity of yellow paint in the bucket = 5/8 x π₯ = 5π₯/8
One bucket of yellow paint is added to the mixture.
New quantity of red paint in the mixture = 3π₯/8
New quantity of yellow paint in the mixture = = 5π₯/8 + π₯ = 13π₯/8
New ratio of red paint and yellow paint in the mixture = 3π₯/8 : 13π₯/8 = 3 : 13
New ratio = 3 : 13.
Unit Conversions (Reference)
Length:
1 m = 3.281 ft
Area:
1 m² = 10.764 ft²,
1 acre = 43,560 ft²,
1 hectare = 2.471 acres
Volume:
1 L = 1000 mL = 1000 cc
Temperature: \[ °F = \frac{9}{5} \times °C + 32, \quad °C = \frac{5}{9} \times (°F - 32) \]
Figure it Out Page 176-177
1. 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
Solution:
Quantity of orange juice = 600ml
Quantity of apple juice = 900ml
Ratio of orange juice to apple juice = 600: 900
Ratio in the simplest form = 600/900= 2/3=2:3
Orange juice : apple juice = 600 : 900 → simplest = 2 : 3.
2. 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?
Solution:
Buses: 162 people → 3 buses (54 each).
Number of buses for 162 students and teachers =3
The buses were full, the capacity of 1 bus = 162/3 = 54
Ratio of number of seats to the number of buses is 54: 1.
54:1 = 2(54): 2(1) = 108 : 2
54:1 = 3(54): 3(1)= 162: 3
54:1 = 4(54): 4(1)=216:4
Capacity of 4 buses = 216
For 204 students, we shall need 4 buses.
Since 216 – 204 = 12,
therefore 12 vacant seats in the buses.
For 204 people → 4 buses needed (12 vacant seats).
3. 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?
Delhi vs Mumbai crowding:
Solution:
Ratio of area to population for Delhi = 1,484:30
let density of Delhi and Mumbai be same and there be x people in Mumbai.
The ratio 1,484 : 30 and 550 : x are in proportion.
1484/30= 550/π₯
1,484x= 30 x 550 = 16,500
X = 16500/1484
x=1,484 = 11.118
There should be 11.118 million people in Mumbai. But population of Mumbai is 20 million. Mumbai is more crowded than Delhi.
Another Method
Area of Delhi = 1,484 sq. km
Population of Delhi = 30 million
Area of Mumbai = 550 sq. km
Population of Mumbai = 20 million
Ratio of area to population for Delhi = 1,484:30
Ratio of area to population for Mumbai = 550: 20
Factor of change of area = 550/1484 0.371 (approx)
Factor of change of population = 20/30=0.667 (approx)
Since 0.667 > 0.371, Mumbai is more crowded than Delhi.
Mumbai more crowded because population density higher.
4. 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?
Crane neck : body = 4 : 6.
Solution:
Ratio of height of neck and height of rest of body of a crane is 4:6
My height is 65 inches = 165 cm. ( 1 inch = 2.54cm) (65 x 2.54 = 165.1 cm)
let the ratio of height of my neck and height of rest of my body be also 4:6.
Height of my neck = ( 4/(4+6) x 165 ) cm
= 660/10 = 66cm
If my height = 165 cm, neck height = \( \frac{4}{10} \times 165 = 66 \) cm.
5. 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 π/π palas of saffron costs π/π niskas, O expert businessman! tell me quickly what quantity of saffron can be bought for 9 niskas?”
Ancient problem:
Solution:
unit of weight in palas and unit of money is niskas. Cost of 2 π/π palas of saffron = π/π niskas
Ratio of weight to price is 2 π/π : π/π or 5/2 x 14 : π/π x 14 = 35:6
let x palas of saffron be bought for 9 niskas.
Ratio of weight to price is x : 9.
These ratios are in proportion.
35 : 6:: x : 9
35/6= (π₯ )/9
6x = 35 x 9
x = 315/6 = 52.5
52.5 palas of saffron can be bought for 9 niskas
\( 2 \frac{1}{2} \) palas saffron → \( \frac{3}{7} \) niskas.
9 niskas → \( \frac{35}{6} \times 9 = 52.5 \) palas saffron.
6. 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?
Harmain age 1, brother 5. When ratio 1 : 2?
Solution:
Age of Harmain and her brother are 1 year and 5 years.
let after x years, the ratio of their ages be 1:2
Age of Harmain after x years = (1 + x) years.
Age of her brother after x years = (5 + x) years After x years, ratio of their ages = 1 + x : 5 + x
The ratios are in proportion. ⇒ 1:2:: 1 + x : 5 + x
1/2=(1+π₯)/(5+π₯)
5 + x = 2(1 + x)
5+x = 2+2x
2x - x = 5 - 2
x = 3
After 3 years, age of Harmain = 1+3 = 4 years.
After 3 years → ages 4 and 8 → ratio 1 : 2.
7. 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?
Gold : water mass ratio = 37 : 2.
Solution:
The ratio of masses of gold and water, when their volumes are same, is 37: 2.
Mass of 1 litre of water = 1kg
let mass of 1 litre of gold = x kg
With equal volumes, ratio of masses of gold and water is x :1.
These ratios are in proportion.
37: 2 :: 1: x
37/2= π₯/1
Thus, the mass of 1 litre of gold is 37/2 kg
1 litre water = 1 kg → 1 litre gold = \( \frac{37}{2} = 18.5 \) kg.
8. 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).
Manure for farming:
Solution: 1 ton = 1000kg
10 tonnes 10×1,000 = 10,000 kg
1 acre = 43,560 sq. ft.
Ratio of cow manure to area of land in kg and sq. ft =10,000 : 43,560 Size of plot = 200 ft. by 500 ft.
Area of plot = 200 x 500 = 1,00,000 sq. ft.
let cow manure required be x kg.
Ratio of cow manure to area of plot = x : 100000
These ratios are in proportion.
10,000 : 43,560 :: x : 1,00,000
10000/43560= π₯/100000
43, 560x =10,000 x 1,00,000 = 1,00,00,00,000
x = 1,00,00,00,000/43560
Required cow manure = 22956.84 kg = 22.95684 tonnes
Plot = 200 ft × 500 ft = 100,000 sq ft.
Manure needed ≈ 22.96 tonnes.
9. 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?
Tap fills 500 ml mug in 15 seconds.
Solution:
Time taken by tap for 500 ml water = 15 seconds
Ratio of volume to time = 500:15 (1 litre = 1,000 ml)
10 litre = 10 x 1,000= 10,000 ml
let time taken to fill a bucket of 10,000 ml be x seconds.
Ratio of volume to time = 10,000 : x
These ratios are proportional.
500 :15 ::10,000 : x
500/15= 10000/π₯
500x = 1,50,000
X = 1500000/500 = 300
Time to fill bucket = 300 seconds = 300/60 = 5 minutes
10 L bucket → 300 seconds = 5 minutes.
10. One acre of land costs ₹15,00,000. What is the cost of 2,400 square feet of the same land?
Land cost: 1 acre = 43,560 sq ft → ₹15,00,000.
Solution: 1 acre = 43,560 square feet.
Cost of 43,560 sq. ft. land = ₹15,00,000
Ratio of area of land to cost = 43,560: 15,00,000
let cost of 2,400 sq. ft. land be ₹x.
Ratio of area of land to cost = 2,400 : x
These ratios are proportional.
43,560 :15,00,000 :: 2,400 : x
43560/1500000= 2400/π₯
43,560x = 2,400 × 15,00,000
x= (2400 π₯ 1500000)/43560 = 82,664.63
Cost of land = ₹82,664.63.
2,400 sq ft → ₹82,664.63.
11. 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?
Tractor vs oxen ploughing:
Solution :
Ratio of efficiency of a tractor to a pair of oxen = 4:1
Time taken by a pair of oxen to plough 1 acre field = 6 hours
Time taken by a tractor to plough
1 acre field = 6/4 = 1.5 hours
Time taken by a pair of oxen to plough 20 acre field = 20×6=120 hours
Time taken by a tractor to plough 20 acre field = 20×1.5 = 30 hours
Oxen: 20 acres × 6 hours = 120 hours.
Tractor (4× faster) → 30 hours.
12. 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?
₹10 coin: copper : nickel = 3 : 1.
Solution:
Ratio of copper and nickel in ₹10 coin = 3:1
Mass of one 10 coin = 7.74 grams
Mass of copper in one ₹10 coin
=3/(3+1) x 7.74 = 3/4 x7.74 = 5.805 grams
Mass of nickel in one ₹10 coin
= 1/(3+1) x 7.74 = 1/4 x7.74 = = 1.935 grams
Cost of 1 kg copper = ₹906
Cost of 1000 grams copper = ₹906
Cost of 5.805 grams copper = 906/1000 x 5.805 = ₹5.26
Cost of 1 kg nickel =₹1341
Cost of 1000 grams nickel =₹1341
Cost of 1.935 grams nickel = 1341/1000 x 1.935 = ₹2.59
In one ₹10 coin, cost of copper and cost of nickel are respectively ₹5.26 and ₹2.59.
Mass = 7.74 g → copper = 5.805 g, nickel = 1.935 g.
Cost: copper = ₹5.26, nickel = ₹2.59.
