Ratio and Proportion are mathematical concepts used to compare quantities. A Ratio compares two or more quantities using division, while a Proportion shows that two ratios are equal.
These concepts are widely used in mathematics, science, engineering, finance, business, map reading, cooking, and many everyday situations involving comparisons and scaling.
A Ratio is a comparison of two or more quantities by division. It shows how many times one quantity contains or is contained in another.
A ratio can be written in any of the following forms:
a : b
a to b
a b
where:
Both quantities must be expressed in the same units before comparing them.
(i) The ratio of 8 boys to 12 girls is 2 : 3.
(ii) The ratio of 15 cm to 25 cm is 3 : 5.
(iii) The ratio of 6 apples to 9 apples is 2 : 3.
A ratio should always be simplified to its lowest terms by dividing both quantities by their Highest Common Factor (HCF).
Simplify the ratio 18 : 24.
Step 1: Write the given ratio.
18 : 24
This is the ratio to be simplified.
Step 2: Find the Highest Common Factor (HCF).
HCF(18,24) = 6
This is the largest number that divides both numbers exactly.
Step 3: Divide both terms by the HCF.
18 ÷ 6 : 24 ÷ 6
18 6 : 24 6
3 : 4
Both quantities have now been simplified.
3 : 4
Converting a ratio to a percentage can be done in two different ways depending on whether you are working with a basic numerical fraction ratio or a real-world part-to-part ratio scenario.
1. Basic Fraction Ratio Formula:
When a ratio is given as a straight fraction relationship (part out of whole), we convert it directly using this formula:
Percentage = a b × 100%
Basic Example: Convert the ratio 3:5 directly into a percentage.
Solution:
Step 1: Identify the values.
a = 3
b = 5
Step 2: Substitute the values into the formula.
Percentage = 3 5 × 100%
Step 3: Simplify.
Percentage = 0.6 × 100%
Percentage = 60%
2. Real-World Part-to-Part Ratio Formula:
When a ratio compares two separate independent parts, you must add the parts together to find the total whole value before calculating the percentage:
Percentage of Part (a) = Part (a) Total Parts (a + b) × 100%
Real-World Example: A school club has a boys-to-girls ratio of 3:5. Find the percentage of the club members who are boys.
Solution:
Step 1: Find the total number of parts in the group.
Boys Share (a) = 3 parts
Girls Share (b) = 5 parts
Total Parts = a + b = 3 + 5 = 8 parts
Step 2: Substitute the values into the real-world ratio formula.
Percentage of Boys = 3 8 × 100%
Step 3: Simplify.
Percentage of Boys = 0.375 × 100%
Percentage of Boys = 37.5%
If we want to find the percentage of girls in the same club, we can do it using two different methods:
Method 1: Subtraction from the Total (100%)
Since the total percentage of the entire club must equal 100%, we can simply subtract the boys' percentage from 100%:
Percentage of Girls = 100% − Percentage of Boys
Percentage of Girls = 100% − 37.5%
Percentage of Girls = 62.5%
Method 2: Using the Formula
Alternatively, we can compute it directly by placing the girls' parts over the total combined parts:
Percentage of Girls = Girls Parts (b) Total Parts (a + b) × 100%
Substitute the values into the alternative formula:
Percentage of Girls = 5 8 × 100%
Simplify the fraction:
Percentage of Girls = 0.625 × 100%
Percentage of Girls = 62.5%
Percentage of Boys = 37.5%
Percentage of Girls = 62.5%
Ratios are not limited to just two numbers like a : b. Students learn to compare three or more quantities simultaneously. This is called a continued ratio and is written in the form a : b : c.
A basic structural example of a three-part ratio is a standard construction concrete mix, which requires cement, sand, and gravel to be mixed together in a strict ratio of 1 : 2 : 4.
Solution:
Step 1: Identify the shared linking term.
The variable B appears in both given ratios. However, its value is 3 in the first ratio and 4 in the second ratio.
A : B = 2 : 3
B : C = 4 : 5
Step 2: Find the Lowest Common Multiple (LCM) for the shared term B.
Find the smallest number that both 3 and 4 can divide into perfectly:
LCM(3, 4) = 12
Step 3: Scale both individual ratios up so that B equals 12 on both sides.
Multiply both terms of the first ratio (A : B) by 4:
(2 × 4) : (3 × 4) = 8 : 12
Multiply both terms of the second ratio (B : C) by 3:
(4 × 3) : (5 × 3) = 12 : 15
Step 4: Merge the scaled values into a single continuous ratio expression.
Since the value for B matches seamlessly across both sets, join them together directly:
A : B : C = 8 : 12 : 15
Final Answer: The combined three-part ratio is 8 : 12 : 15.
A Proportion is a statement showing that two ratios are equal.
a : b = c : d
or
a b = c d
where:
If two ratios are equal, they form a proportion.
(i) 2 : 3 = 4 : 6
(ii) 5 : 8 = 10 : 16
(iii) 7 : 9 = 14 : 18
A ratio compares quantities, while a proportion states that two ratios are equal.
For example,
3 : 5 is a ratio.
3 : 5 = 12 : 20 is a proportion because both ratios are equal.
Proportions are commonly solved using cross multiplication.
Find the value of x if the ratios 4 : 7 and 12 : x form a proportion.
Solution:
Step 1: Set up the proportion as a fraction equation.
4 7 = 12 x
The known ratios are set equal to each other.
Step 2: Cross multiply to solve for x.
Multiply the numerator of each side by the denominator of the opposite side:
4 × x = 12 × 7
4x = 84
Step 3: Isolate x by dividing both sides by 4.
x = 84 4
x = 21
This balances the proportion correctly.
x = 21
You can add a quick check at the end to prove x = 21 is correct by simplifying the second ratio using your answer:
Since 4 7 = 4 7 , the answer is verified.
Students must learn that ratios do not always change in the same direction. There are two primary types of proportion used to solve real-world problems:
In a Direct Proportion, as one quantity increases, the other quantity increases at the exact same rate. If one quantity decreases, the other decreases too.
Formula:
Where k is a fixed number called the constant of proportionality.
Solution:
Step 1: Set up the direct proportion as a fraction equation.
Let the unknown cost of 5 books be y. Since more books mean a higher cost, set their ratios equal to each other:
Step 2: Cross multiply to remove the denominators.
Multiply the top of each side by the bottom of the opposite side:
3 × y = 24 × 5
3y = 120
Step 3: Isolate y by dividing both sides.
Divide 120 by 3 to find the final price:
y = 120 ÷ 3
y = 40
Final Answer: The cost of 5 text books is $40.
In an Inverse (Indirect) Proportion, as one quantity increases, the other quantity decreases at the exact same rate. If one quantity decreases, the other increases.
Formula:
x × y = k
Where k is the constant product of the two interacting quantities.
Solution:
Step 1: Find the constant work value (k).
More workers mean it will take fewer days. Multiply the known worker and day values together to find the total work required:
4 workers × 6 days = 24 work units (k = 24)
Step 2: Set up the equation for the new number of workers.
Let the unknown number of days be d. The product of 8 workers and d days must equal the same constant work value:
8 × d = 24
Step 3: Isolate d using division.
Divide the total work units by the new number of workers:
d = 24 ÷ 8
d = 3
Final Answer: It will take 8 workers exactly 3 days to build the wall.
If three quantities are arranged in a continuous or chained proportion line such that the second term repeats, it creates a unique geometric relationship.
Form:
Where:
Formula :
Mean Proportional Example:
Find the mean proportional between the values 4 and 9.
Solution:
Step 1: Set up the continuous proportion format using variables.
Let the unknown middle mean variable be b, positioning it on both standard inner terms:
Step 2: Cross multiply to establish the squared variable value.
Multiply the diagonal matching terms together directly:
b × b = 4 × 9
b2 = 36
Step 3: Isolate b by taking the square root of both sides.
Extract the square root value to determine the baseline variable level:
b = √36
b = 6
Final Answer: The mean proportional between 4 and 9 is 6.
Solution:
Step 1: Set up the continuous proportion format using variables.
Let the unknown third variable be c. Position the first term as 4, and repeat the second term 6 as the mean value:
Step 2: Cross multiply to remove the denominators.
Multiply across the diagonals to create your whole-number equation structure:
4 × c = 6 × 6
4c = 36
Step 3: Isolate c by dividing both sides.
Divide 36 by 4 to get the final evaluation step outcome:
c = 36 ÷ 4
c = 9
Final Answer: The third proportional to 4 and 6 is 9.
When three distinct terms a, b, and c are known in the proportion a : b = c : d, the missing final value d is formally called the Fourth Proportional.
Solution:
Step 1: Map the given values into the proportion fraction formula.
Assign a = 2, b = 5, c = 6, and let d remain the unknown variable setup:
Step 2: Cross multiply to remove the denominators.
Multiply the numerator of each side by the denominator of the opposite side:
2 × d = 5 × 6
2d = 30
Step 3: Isolate d by dividing both sides.
Divide 30 by 2 to balance out the equation statement:
d = 30 ÷ 2
d = 15
Final Answer: The fourth proportional to 2, 5, and 6 is 15.
When dealing with complex fractional equations, students learn standard transformation theorems to manipulate proportions quickly without fully cross-multiplying. If we assume that:
Then the following properties are always mathematically true:
Students bridge the gap between basic arithmetic and graphing functions by learning about unit scales.
Solution:
Step 1: Set up the baseline ratio fraction.
Step 2: Divide both terms by the denominator to drop the bottom value to 1.
Calculate the units handled in a single baseline minute marker:
120 ÷ 4 = 30 juice boxes
4 ÷ 4 = 1 minute
Step 3: State the final unit rate and constant value.
The unit rate is 30 juice boxes per minute.
The constant of proportionality (k) is 30.
Final Answer: The machine operates at a unit rate of 30 juice boxes/minute, where k = 30.
Attempt the following Questions:
(i) A school has 360 boys and 540 girls. Express the ratio of boys to girls in its simplest form and determine the percentage of students who are boys.
(ii) The ratio of red, blue and green marbles in a container is 5 : 7 : 8. If there are 160 marbles altogether, calculate the number of red, blue and green marbles.
(iii) A recipe requires flour and sugar in the ratio 4 : 3. If 900 g of sugar is used, determine the amount of flour required.
(iv) Find the value of x if the ratios 8 : 15 and x : 45 form a proportion.
(v) Twelve workers can complete a construction project in 15 days. Assuming all workers have equal efficiency, how many days will 20 workers take to complete the same project?
b2 = ac
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