Students often refer to Class 8 Maths Notes and Chapter 7 Proportional Reasoning 1 Class 8 Notes during last-minute revisions.
Class 8 Maths Chapter 7 Notes Proportional Reasoning 1
Class 8 Maths Notes Chapter 7 – Class 8 Proportional Reasoning 1 Notes
→ Ratios in the form of a : b indicate that for every ‘a’ unit of the first quantity, there are ‘b’ units of the second quantity. ‘a’ and ‘b’ are the terms in the ratio.
→ Two ratios a : b and c : d are proportional (written a : b :: c : d) if their terms change by the same factor, i.e., if ad = bc.
→ If x is divided into two parts in the ratio m : n, then the quantity of the first part is m × \(\frac{x}{m+n}\) and the quantity of the second part is n × \(\frac{x}{m+n}\).
→ Ratio: If a and b are two quantities of the same kind and with the same units, such that b is not equal to 0, then the quotient \(\frac {a}{b}\) is called the ratio between a and b. The ratio \(\frac {a}{b}\) has no unit and it is written as a : b.
→ Ratio in the simplest form: If HCF of a and b is d, then the ratio \(\frac{a}{d}: \frac{b}{d}\) is called the ratio in the simplest form of the ratio a : b.
→ Proportion: If two ratios in their simplest form are the same, then the ratios are said to be in proportion.
→ If a : b :: c : d, then we have
- \(\frac{a}{b}: \frac{c}{d}\)
- Factor of change of a = Factor of change of b
- Product of extremes terms = Product of middle terms, i.e., ad = bc
→ Trairasika-The Rule of Three: If two ratios are proportional involving an unknown quantity, then the unknown quantity can be found by using cross multiplication.
→ Rule of Sharing: If quantity x is to be divided in two parts in the ratio m : n, then the two parts are respectively \(\left(\frac{m}{m+n}\right) x\) and \(\left(\frac{n}{m+n}\right) x\).
→ Unit Conversions
Length:
- 1 m = 100 cm
- 1 m = 3.281 ft
Area:
- 1 sq.m = 10,000 sq.cm
- 1 sq.m = 10.765 sq.ft
- 1 acre = 4,046.856 sq.m
- 1 acre = 4,840 sq.gaj
- 1 acre = 43,560 sq ft
- 1 hectare = 10,000 sq.m
Volume:
- 1 mL = 1 cc
- 1 litre = 1,000 mL or 1,000 cc
Temperature:
- Fahrenheit = \(\frac {9}{5}\) × Celsius + 32
Observing Similarity in Change
Consider the following four rectangles:
Rectangle IV is also a square.
We observe the following:
(i) Rectangles I and II have different sizes, but they look similar, because the width of each rectangle is double the height of the rectangle.
We can also discuss the similarity in terms of the ‘factor of change’.
Here \(\frac{70}{50}=\frac{7}{5}\) so 70 = \(\frac {7}{5}\) × 50
∴ The width of rectangle II has changed by the factor \(\frac {7}{5}\).
Also, \(\frac {7}{5}\) × 25 = 35, so the height of rectangle II has also changed by the same factor \(\frac {7}{5}\).
Since, width and height of rectangle II have changed by the same factor (=\(\frac {7}{5}\)), the rectangles I and II are similar.
(ii) Rectangle III is not similar to rectangle I, because the width of rectangle I is double its height, whereas in rectangle III, the width is one and a half times its height.
Using ‘factor of change’
Here, factor of change of width = \(\frac{60}{50}=\frac{6}{5}\)
and factor of change of height = \(\frac{40}{25}=\frac{8}{5}\)
Since \(\frac{6}{5} \neq \frac{8}{5}\), rectangle III is not similar to rectangle I.
Similarly, rectangle III is not similar to rectangle II.
(iii) Rectangle IV is not similar to rectangle I, because the width of rectangle I is double its height, whereas in rectangle IV, the width is equal to its height.
Using ‘factor of change’
Here, factor of change of width = \(\frac{45}{50}=\frac{9}{10}\)
and factor of change of height = \(\frac{45}{25}=\frac{9}{5}\)
Since \(\frac{45}{50}=\frac{9}{10}\), rectangle IV is not similar to rectangle I.
Similarly, rectangle IV is not similar to rectangle II.
(iv) Rectangle IV is not similar to rectangle III, because the width of rectangle III is one and a half times its height, whereas in rectangle IV, the width is equal to its height.
Using ‘factor of change’
Here, factor of change of width = \(\frac{45}{60}=\frac{3}{4}\)
and factor of change of height = \(\frac{45}{40}=\frac{9}{8}\)
Since \(\frac{3}{4} \neq \frac{9}{8}\), the rectangle IV is not similar to rectangle III.
Ratios
If a and b are two quantities of the same kind and with the same units, such that b is not equal to 0, then the quotient \(\frac {a}{b}\) is called the ratio between a and b.
The ratio \(\frac {a}{b}\) has no unit and it is written as a : b.
In the ratio a : b, the number a is called the antecedent, and the number b is called the consequent.
In the 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.
Two quantities can be compared only if they are in the same unit.
Ratios in Their Simplest Form
We know that \(\frac {30}{40}\) is equal to \(\frac {3}{4}\).
∴ The ratio 30 : 40 is equal to the ratio 3 : 4. The ratio 3 :4 is simpler than the ratio 30 : 40.
Let a : b be a ratio.
Let d be the HCF of numbers a and b.
The ratio \(\frac{a}{d}: \frac{b}{d}\) is called the ratio in the simplest form of the ratio a : b.
For example, consider the ratio 105 : 84.
The HCF of 105 and 84 is 21.
∴ The simplest form of the ratio 105 : 84 is \(\frac{105}{21}: \frac{84}{21}\), i.e., 5 : 4.
Example 1.
Let the length of a room be 30 m and its breadth be 20 m.
The ratio of length of the room to the breadth of the room in the simplest form = \(\frac{30}{20}=\frac{3}{2}\) = 3 : 2
Example 2.
The ratios 12 : 9, 20 : 15 and 64 : 48 in the simplest form are equal to 4 : 3, because \(\frac{12}{9}=\frac{20}{15}=\frac{64}{48}=\frac{4}{3}\)
Example 3.
The ratio in the simplest form of 2 m and 50 cm is equal to \(\frac{200 \mathrm{~cm}}{50 \mathrm{~cm}}=\frac{4}{1}\) = 4 : 1
1. Proportion
If two ratios in their simplest form are the same, we say that the ratios are in proportion.
Let the ratios a : b and c : d be in proportion.
Let a’ : b’ and c’ : d’ be the simplest forms of the ratios a : b and c : d, respectively.
∴ \(\frac{a}{b}=\frac{a^{\prime}}{b^{\prime}}\) and \(\frac{c}{d}=\frac{c^{\prime}}{d^{\prime}}\)
Since ratios a : b and c : d are in proportions, we have
\(\frac{a^{\prime}}{b^{\prime}}=\frac{c^{\prime}}{d^{\prime}}\)
∴ \(\frac{a}{b}=\frac{c}{d}\) (∵ \(\frac{a^{\prime}}{b^{\prime}}=\frac{a}{b}\) and \(\frac{c^{\prime}}{d^{\prime}}=\frac{c}{d}\))
∴ If ratios a : b and c : d are in proportion, then we have \(\frac{a}{b}=\frac{c}{d}\).
2. Notation
If the ratios a : b and c : d are in proportion, then we write a : b :: c : d to indicate that the ratios a : b and c : d are in proportion.
For example, ratios 10 : 25 and 34 : 85 are in proportion, because \(\frac{10}{25}=\frac{5 \times 2}{5 \times 5}=\frac{2}{5}\) and \(\frac{34}{85}=\frac{17 \times 2}{17 \times 5}=\frac{2}{5}\).
85 “ 17 x 5 _ 5 ’
∴ 10 : 25 :: 34 : 85.
If two ratios are in proportion, then we also say that these ratios are proportional.
If the ratios a : b and c : d are in proportion, then we have \(\frac{a}{b}=\frac{c}{d}\).
∴ ad = bc
∴ If the ratios a : b and c : d are in proportion, then we have:
Product of extremes terms = Product of middle terms.
Sharing, But Not Equally!
Suppose we have 12 similar coins, and these coins are to be shared by A and B.
If A and B get 6 coins each, then we say that the coins are divided in the ratio 6 : 6 or simply 1 : 1.
If A gets 5 coins, then B will get 12 – 5 = 7 coins.
In this case, we say that the coins are divided in the ratio 5 : 7.
Conversely, let 12 coins are to be divided among A and B in the ratio 3 : 1.
It means that if we have 4 coins, then A gets 3 coins and B gets 1 coin.
In other words, for every 4 coins, the shares of A and B are 3 and 1, respectively.
In 12 coins, we can form 3 groups of 4 coins each.
∴ Out of 12 coins, i.e., out of 3 groups of 4 coins, A will get 3 × 3 = 9 coins and B will get 3 × 1 = 3 coins.
Equivalently, we say that the 12 coins are divided into two parts \(\left(\frac{3}{3+1}\right)\)12 = 9 coins and \(\left(\frac{1}{3+1}\right)\)12 = 3 coins.
In general, suppose we are to divide a quantity x in the ratio m : n.
The ratio is m : n.
The sum of ratios is m + n.
∴ The quantity m + n is divided into two parts, m and n.
∴ The quantity x is divided into two parts \(\left(\frac{m}{m+n}\right)\)x and \(\left(\frac{n}{m+n}\right)\)x.
∴ If quantity x is to be divided into two parts in the ratio m : n, the two parts are \(\left(\frac{m}{m+n}\right)\)x and \(\left(\frac{n}{m+n}\right)\)x respectively.
Unit Conversions
We know that in the study of ratios, we always assume that the units of a and b in the ratio a : b are always the same.
Sometimes, we need the conversion of units in discussing problems involving ratios.
For example, the ratio of 4 mm to 2 cm is 4 : 20 and not 4 : 2.
The following are some important formulae for unit conversions.
Length:
- 1 metre = 100 cm
- 1 metre = 3.281 ft.
Area:
- 1 square metre = 10,000 square cm
- 1 square metre = 10.765 square ft.
- 1 acre = 4046.856 square m.
- 1 acre = 4,840 square gaj
- 1 acre = 43,560 square ft.
- 1 hectare = 10,000 square m.
Volume:
- 1 millilitre (mL) = 1 cubic cm (cc)
- 1 litre = 1,000 mL or 1,000 cc
Temperature:
- Fahrenheit = \(\frac {9}{5}\) × Celsius + 32.
Observing Similarity in Change Class 8 Notes
Shapes whose dimensions change by the same scale factor look similar.
Example Two rectangles 8 cm × 5 cm and 4 cm × 2.5 cm are similar.
Scale factor = 0.5
Different ways of comparing numbers are:
(a) Comparison by Difference
Example:
Adi is 4 years older than Bhavya.
Mehul has ₹ 20 less than Naina.
(b) Comparison by Fraction
Example:
Suman has twice the chocolates Aaliya has.
Neetu has one-third the number of balloons as Amaan has.
(c) Comparison by Ratio: a : b read as “a ratio b” or ‘a is to b’
For every a items of first type, there will be b items of second type.
Example: For every 2 pencils Javed has, he has 1 eraser.
So, Pencils : Erasers = 2:1
When ratio is reduced to lowest terms, it is said to be in simplest form.
Example-. 18 : 24 in lowest form is 3 : 4. [as (18 ÷ 6) : (24 ÷ 6)]
- Ratio has no units.
- First term of a ratio is called antecedent.
- Second term of a ratio is called consequent.
We are all familiar with digital images. We often change the size and orientation of these images to suit our needs. Observe the set of images below.
We can see that all the images are of different sizes.
Which images look similar and which ones look different?
Images (A, C, and D) look similar, even though they have different sizes.
Do images B and E look like the other three images?
No, they are slightly distorted. The tiger appears elongated in B, and compressed and fatter in E!
Why? You may notice that images A, C, and D are rectangular, but E is square. Maybe that is why E looks different. But B is also a rectangle!
Why does it look different from the other rectangular images?
Can we observe any pattern to answer this question? Perhaps by measuring the rectangles?
What makes images A, C, and D appear similar, and B and E different?
When we compare image A with C, we notice that the width of C is half that of A. The height is also half of A. Both the width and height have changed by the same factor (through multiplication), \(\frac {1}{2}\) in this case. Since the widths and heights have changed by the same factor, the images look similar.
When we compare image A with image B, we notice that the width of B is 20 millimeters (mm) less than that of A. The height is 20 mm less than the height of A. Even though the difference (through subtraction) is the same, the images look different. Have the width and height changed by the same factor? The height of B is half the height of A. But the width of B is not half the width of A. Since the width and height have not changed by the same factor, the images look different.
Can you check by what factors the width and height of image D change as compared to image A? Are the factors the same?
Images A, C, and D look similar because their widths and heights have changed by the same factor. We say that the changes to their widths and heights are proportional.
Ratios Class 8 Notes
- When two ratios have the same simplest form, they are said to be in proportion.
- Symbol used for proportion is
- If a, b, c, d are in proportion, it implies a : b = c : d or a : b :: c : d.
- First and last terms i.e., a and d are called extremes (extreme terms).
- Second and third terms i.e., b and c are called means (mean terms).
- Product of Extremes = Product of Means.
We use the notion of a ratio to represent such proportional relationships in mathematics.
We can say that the ratio of the width to the height of image A is 60 : 40.
The numbers 60 and 40 are called the terms of the ratio.
The ratio of width to height of image C is 30 : 20, and that of image D is 90 : 60.
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.
So, in image A, we can say that for every 60 mm of width, there are 40 mm of height.
We can say that the ratios of width to height of images A, C, and D are proportional because the terms of these ratios change by the same factor. Let us see how.
Image A – 60 : 40
Multiplying both the terms by \(\frac {1}{2}\), we get
60 × \(\frac {1}{2}\) : 40 × \(\frac {1}{2}\)
which is 30 : 20, the ratio of width to height in image C.
By what factor should we multiply the ratio 60 : 40 (image A) to get 90 : 60 (image D)?
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.
Ratios in Their Simplest Form Class 8 Notes
We can reduce ratios to their simplest form by dividing the terms by their HCF.
In image A, the terms are 60 and 40.
What is the HCF of 60 and 40?
It is 20. Dividing the terms by 20, we get the ratio of image A to be 3 : 2 in its simplest form.
The ratio of image D is 90 : 60. Dividing both terms by 30 (HCF of 90 and 60), we get the simplest form to be 3 : 2.
So the ratios of images A and D are proportional as well.
What is the simplest form of the ratios of images B and E?
The ratio of image B is 40 : 20; in its simplest form, it is 2 : 1.
The ratio of image E is 60 : 60; in its simplest form, it is 1 : 1.
These ratios are not the same as 3 : 2. So, we can say that the ratios of width to height of images B and E are not proportional to the ratios of images A, C, and D.
When two ratios are the same in their simplest forms, we say that the ratios are in proportion, or that the ratios are proportional.
We use the ‘::’ symbol to indicate that they are proportional.
So a : b :: c : d indicates that the ratios a : b and c : d are proportional.
Thus, 60 : 40 :: 30 : 20 and 60 : 40 :: 90 : 60.
Problem Solving with Proportional Reasoning Class 8 Notes
If A gets m articles and B gets x articles, then total articles is m + n.
- A gets m out of m + n articles.
OR
A’s share is \(\frac{m}{m+n}\) - B gets n articles out of m + n articles.
OR
B’s share is \(\frac{n}{m+n}\) - If P articles are to be divided between A and B in the ratio
A’s share = \(\frac{m}{m+n}\) × P and
B’s share = \(\frac{n}{m+n}\) × P
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.
What is the HCF of 72 and 96?
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?
To maintain the same sweetness, the ratio of the number of glasses of lemonade to the number of spoons of sugar should be proportional.
For 6 glasses of lemonade, she added 10 spoons of sugar.
The ratio of glasses of lemonade to spoons of sugar is 6 : 10.
If she needs to make 18 more glasses of lemonade, how many spoons of sugar should she use?
We can model this problem as 6 : 10 :: 18 : ?
We know that each term in the ratio must change by the same factor for the ratios to be proportional.
How can we find the factor of change in the ratio?
The first term has increased from 6 to 18.
To find the factor of change, we can divide 18 by 6 to get 3.
The second term should also change by the same factor.
When 10 increases by a factor of 3, it becomes 30.
Thus, 6 : 10 :: 18 : 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?
In Nitin and Hari’s case, we should compare the ratio of the length of the wall to the bags of cement used by each of them and see whether they are proportional.
The ratio in Nitin’s case is 60 : 3, i.e., 20 : 1 (in its simplest form).
The ratio in Hari’s case is 40 : 2, i.e., 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? Is the teacher-to-student ratio in your school proportional to the one in my school?
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?
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 (i.e., 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 : ______
In the first ratio, we don’t know the first term.
But the second term is 42.
It is 2 times the second term of the ratio 14 : 21.
So, the first term should also be 2 times 14 (the first term).
Hence the proportional ratio is 28 : 42.
For the second ratio, the first term is 6.
What factor should we multiply 14 by to get 6? Can it be an integer? Or should it be a fraction?
We can model this as 14y = 6.
So, y = \(\frac {6}{14}\) = \(\frac {3}{7}\)
So, we need to multiply 21 (the second term of 14 : 21) also by the same factor \(\frac {3}{7}\).
21 × \(\frac {3}{7}\) is 9. So, the ratio is 6 : 9.
In the third ratio, the first term is 2.
We can see that when we divide 14 (the first term of 14 : 21) by 7 (HCF of 14 and 21) we get 2.
If we divide 21 by 7, we get 3.
So, the ratio is 2 : 3.
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 the decoction with 30 mL of milk.
The ratio here is 20 : 30.
Why is this coffee stronger?
And when they want ‘lighter’ filtered 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.
Trairasika – The Rule of Three Class 8 Notes
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?
We can find that by dividing the terms \(\frac{80}{120}=\frac{2}{3}\).
The number of students is reduced by a factor of \(\frac {2}{3}\).
On multiplying the weight of rice by the same factor,
we get, 15 × \(\frac {2}{3}\) = 10
So, the cook should make 10 kg of rice on that day.
The situation above is a typical example of a problem where we need to use proportional reasoning to find a solution.
Four quantities are linked proportionally, out of which three are known, and we must find the fourth, unknown quantity.
To solve such problems, we can model two proportional ratios using algebraic notation as a : b :: c : d.
For these two ratios to be proportional, we know that term c should be a multiple of term a by a factor, say f, and term d should be a multiple of term b by the same factor f.
So, c = fa ……(1)
d = fb ……(2)
From (1) and (2), we can say that,
f = \(\frac {c}{a}\) and f = \(\frac {d}{b}\)
Therefore, \(\frac{c}{a}=\frac{d}{b}\)
Multiplying both sides by ab, we get,
ab × \(\frac {c}{a}\) = ab × \(\frac {d}{b}\)
bc = ad or ad = bc
Thus, when a : b :: c : d, then ad = bc.
This is known as cross multiplication of terms.
Since ad = bc, we can show that d = \(\frac {bc}{a}\)
Two ratios are proportional if their terms are equal when cross-multipled. The fourth unknown quantity can be found through such cross multiplication.
In ancient India, Aryabhata (199 CE) and others called such problems of proportionality the Rule of Three problems. There were 3 numbers given the pramana (measure ‘a’ in our case), the phala (fruit ‘b’ in our case), and the ichchha (requisition ‘c’ in our case). To find the ichchhaphala (yield ‘d’ in our case), Aryabhata says, “Multiply the phala by the ichchhā and divide the resulting product by the pramana.”
In other words, Aryabhata says,
“pramana : phala :: ichchha : ichchhaphala,”
therefore, pramana × ichchhaphala = phala × ichchha.
Thus, ichchhaphala = \(\frac{\text { phala } \times \text { ichchha }}{\text { pramana }} \text {. }\)
Using the cross multiplication method proposed by Aryabhata, ancient Indians solved complex problems that involved proportionality.
Example 9: 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.
150 : 90 :: 4 : ?
Is this the right way to formulate the question?
No, because 150 is in minutes, but 4 is in hours.
The second ratio should use the same units for time as the first ratio.
Since 4 hours is 240 minutes, the right form is 150 : 90 :: 240 : ?
How can you find the distance covered in 240 minutes?
Discuss with your classmates and find the answer using different strategies.
We can model this proportion as 150 : 90 :: 240 : x.
By cross multiplication, we get
150 × x = 240 × 90
Therefore, x = \(\frac{240 \times 90}{150}\) = 144
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?
The ratio of weight to price of the Himachal tea is 200 : 200.
What is the weight-to-price ratio of the Meghalaya tea? Is it 1 : 800?
This would not be appropriate, because we considered the weight in grams in the case of Himachal.
So, the weight-to-price ratio is 1000 : 800 in Meghalaya after we convert the weight to grams.
To check if the ratios are proportional, we need to see if both ratios are the same in their simplest forms.
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.
Which tea is more expensive? Why?
To answer the question as to which tea is more expensive, we should compare the price of tea for the same weight in both places.
What is the price of 1 kg of tea from Meghalaya? It is ₹ 800.
In Himachal, if 200 g of tea costs rupees 200, what is the cost of 1 kg of tea?
Let us say that the price of 1 kg of tea is x rupees.
200 g is \(\frac {1}{5}\) of 1 kg.
So, \(\frac {1}{5}\) × x = 200
Multiplying both sides by 5, we get
\(\frac {1}{5}\) × 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.
Activity 1: Take your favourite dish. Find out all the ingredients and their respective quantities needed to make the dish for your family. Suppose you are celebrating a festival and you want to invite 15 guests. Find out the quantities of the ingredients required to cook the same dish for them.
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 decrease when the speed increases. So this problem cannot be modelled as 50 : 2 :: 75 : __
Activity 2: Go to the market and collect the prices of different sizes of shampoo containers of the same shampoo, and create a table like the one given below. See if the volume of shampoo is proportional to the price.
Let us compare the ratios for the sample table above.
The ratio of the volume of a sachet to a small bottle is 6 : 180. The ratio of their prices is 2 : 154. Are these ratios proportional?
Why do you think that the ratio of the prices is not proportional to the ratio of the volumes?
Discuss the pros and cons of different-sized bottles for the company and for customers. For reducing the ecological footprint, what would you recommend to the company and to the customer?
Does the same occur for other products?
Make similar tables for other products in the market, capturing different prices for different measures of the same product, e.g., rice or atta (flour).
Observe the products for which the prices are proportional to the different measures.
Discuss in class the proportionality of prices to measures of the same product.
Sharing, but Not Equally! Class 8 Notes
Activity 3: Form a pair. Collect 12 countable objects or counters (it can be coins, seeds, or pebbles). Now, share them between the two of you in different ways. If you divide them equally, what is the ratio of the number of counters to each of you?
Each of you will get 6 counters. So, the ratio is 6 : 6, or 1 : 1 in its simplest form. Now let us not share equally.
If your partner gets 5 counters, how many objects will you get? What is the ratio of the counters?
The ratio of the counters of your partner to yours is 5 : 7.
Now, if you want to share the counters between the two of you in the ratio of 3 : 1, how many counters would each of you get?
Share the counters in different ways and see which combination is in the ratio 3 : 1.
One way to share the counters in the ratio of 3 : 1 is as follows.
- Your partner takes 3 counters and you take 1 counter. There are now 8 counters left.
- Your partner takes 3 more counters, and you take 1 more counter. There are now 4 counters left.
- Your partner takes 3 more counters, and you take 1 more counter. There are no more counters left.
So, your partner gets 9 counters in total, and you get 3 counters.
When we divide 12 counters in the ratio of 3 : 1 between two people, one gets 9 counters and the other gets 3 counters.
Now, if you want to share 42 counters between the two of you in the ratio of 4 : 3, how will you do it?
Using the same procedure would take a long time! There is a simpler way to share the whole with parts in a specified ratio.
You need to divide 42 into groups such that your partner gets 4 groups and you get 3 groups.
What is the size of each group?
If your partner gets 4 groups and you get 3 groups, the total number of groups is 7.
So, the size of each group is 42 ÷ 7 = 6.
Multiplying the number of groups by the size of each group, your partner gets 24 counters, and you get 18 counters, when you share 42 counters in the ratio of 4 : 3.
In general, when we want to divide a quantity, say x, in the ratio m : n, we do the following:
- We need to split x into groups such that it can be divided into two parts, where the first part has m groups and the second part has n groups.
- But what is the size of each group? This can be found out by dividing x by the number of groups. The number of groups is m + n. So, the size of each group is \(\frac{x}{m+n}\).
- So, the fist part has m × \(\frac{x}{m+n}\) objects and the second part has n × \(\frac{x}{m+n}\) objects.
Thus, if we want to divide a quantity x in the ratio of m: n, then the parts will be m × \(\frac{x}{m+n}\) and n × \(\frac{x}{m+n}\).
We see that m × \(\frac{x}{m+n}\) : n × \(\frac{x}{m+n}\) :: m : n.
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 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 \(\frac{3}{(3+1)}\) × 40 = 30 kg.
The weight of cement is \(\frac{1}{(3+1)}\) × 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 \(\frac {2}{5}\) times the first term.
Since the new ratio is equivalent to 5 : 2, the second term in the new ratio should also be \(\frac {2}{5}\) times of 30.
\(\frac {2}{5}\) × 30 = 12.
The new mixture should have 12 kg of cement if the ratio of sand to cement is to be 5 : 2.
There are 10 kg of cement already. So, we need to add 2 kg of cement to the original mixture.
Unit Conversions Class 8 Notes
We have noticed earlier that solving problems with proportionality often requires us to convert units from one system to another. Here are a few important unit conversions for your reference.
Length:
1 metre = 3.281 feet
Area:
1 square metre = 10.764 square feet
1 acre = 43,560 square feet
1 hectare = 10,000 square metres
1 hectare = 2.471 acres
Volume:
1 millilitre (mL) = 1 cubic centimetre (cc)
1 litre = 1,000 mL or 1,000 cc
Temperature:
Temperature conversion between Fahrenheit and Celsius is a bit more complicated.
0°C = 32°F, and
Fahrenheit = \(\frac {9}{5}\) × Celsius + 32
and Celsius = \(\frac {5}{9}\) × (Fahrenheit – 32)
For example, 25°C is 77°F.