---
**Definition:** A percentage (denoted by the symbol %) is a way of expressing a number as a fraction out of 100. The word "per cent" comes from the Latin phrase "per centum," meaning "by the hundred" or "out of hundred."
**Key Concept:** When we say 25%, it means 25 out of every 100. For example:
**Mathematical Expression:**
When we calculate p% of some quantity s, we use:
p% of s = (p/100) × s
**Example:** 50% of some quantity s means:
50% = (50/100) × s = (1/2) × s
This shows that percentages are simply fractions where the denominator is always 100.
**Important Truth:** Percentages are fractions in disguise. Every percentage can be written as a fraction with denominator 100.
**Examples of Common Conversions:**
There are multiple methods to convert a fraction to its percentage form. Let us understand them through examples.
#### Method 1: Using Equivalent Fractions
**Process:** Find an equivalent fraction where the denominator is 100.
**Example 1: Express 3/4 as a percentage**
Real-life context: Surya mixes red and yellow paint. The red paint makes up 3/4 of the mixture. What percentage is red?
Step 1: Find what we multiply the denominator 4 by to get 100.
100 ÷ 4 = 25
Step 2: Multiply both numerator and denominator by 25:
3/4 = (3 × 25)/(4 × 25) = 75/100
Step 3: Write as a percentage:
75/100 = 75%
**Visual Understanding (Bar Model):**
```
0 1/4 1/2 3/4 1
0% 25% 50% 75% 100%
0 25/100 50/100 75/100 100/100
```
**Answer:** Red paint makes up 75% of the mixture. This means yellow paint makes up 25% (since 100% - 75% = 25%).
#### Method 2: Using Algebraic Approach
**Process:** Set up a proportion and solve for the percentage.
**Example 2: Express 2/5 as a percentage**
Real-life context: Surya wants to save 2/5 of his prize money to buy a new canvas.
Step 1: Set up the equation:
2/5 = x/100
Step 2: Multiply both sides by 100:
x = (2/5) × 100
Step 3: Calculate:
x = (2 × 100)/5 = 200/5 = 40
Step 4: Write the answer:
2/5 = 40/100 = 40%
**Answer:** Surya will save 40% of his prize money.
#### Method 3: Direct Multiplication by 100
**General Rule:** To convert any fraction to a percentage, multiply the fraction by 100.
**Formula:** Fraction = a/b
Percentage = (a/b) × 100%
**Example 3: Express 3/5 as a percentage**
Step 1: Multiply by 100:
(3/5) × 100 = 300/5 = 60%
**Answer:** 3/5 = 60%
**Why This Works:** Since a percentage is defined as "out of 100," we're essentially finding how many parts out of 100 the fraction represents. Multiplying by 100 scales any fraction to this base of 100.
**Important Fact:** Any percentage z% can be written as the fraction z/100. This fraction can then be reduced to simpler equivalent forms.
**Example 4: Express 24% as a fraction**
Step 1: Write the percentage as a fraction:
24% = 24/100
Step 2: Simplify by finding the GCD (Greatest Common Divisor):
GCD of 24 and 100 = 4
Step 3: Reduce the fraction:
24/100 = (24÷4)/(100÷4) = 6/25
**Other equivalent forms of 24/100:**
**Answer:** 24% = 6/25 (in simplest form), and all equivalent fractions are valid representations.
#### The Comparison Problem
Consider this situation: A biscuit factory tests two varieties.
**Question:** Which variety is more sugary?
When fractions have different denominators, comparison is difficult and requires calculation. We would need to find a common denominator or convert to decimals.
**With Percentages:**
**Observation:** It is immediately clear that Variety 2 is more sugary because 28.88% > 26.47%.
#### Why 100 as the Denominator?
The choice of 100 is special because of several advantages:
1. **Base 10 System:** Our number system uses base 10, and 100 fits perfectly with decimals:
2. **Mental Feasibility:** 100 is large enough to provide detail but simple enough to grasp mentally. For comparison:
3. **Historical Use:** The concept of "per hundred" was used in ancient times:
#### Alternative Systems (For Reference)
While 100 is standard, other denominators exist:
Understanding where percentages appear helps appreciate their importance:
1. **Human Body:** The human body is approximately 60% water by weight
2. **Food Science:** Ice cream contains about 30-50% air by volume
3. **Global Events:** 45% of the world's population watched at least part of the 2022 FIFA World Cup
4. **Health Statistics:** Over 80% of teenagers globally fail to meet the recommendation of at least one hour of daily physical activity
5. **Astronomy:** About 99.86% of the Solar System's mass is contained in the Sun
6. **Environment:** An estimated 52% of agricultural land worldwide is degraded
---
**Key Understanding:** When we express a percentage, we are expressing it relative to some whole (or base). When comparing percentages, we must ensure they refer to the same or comparable wholes.
#### The Importance of the Base Quantity
**Example 1: Comparing Sugar Content**
Real-life context: Madhu and Madhav ate different quantities of different biscuits.
**Question:** Who ate more sugar?
**Critical Insight:** We cannot compare just the percentages without knowing the quantities eaten!
**Scenario A: If both ate 100g of biscuits**
**Scenario B: If Madhu ate 120g and Madhav ate 95g**
We need to calculate the actual amount of sugar in each case.
For **Madhu (120g with 25% sugar):**
**Method 1: Using Equivalent Fractions and Proportions**
25g sugar : 100g biscuits = s g sugar : 120g biscuits
25:100 :: s:120
This is a proportional relationship. We set:
25/100 = s/120
Solving for s:
s = (25/100) × 120 = 30g sugar
**Method 2: Direct Calculation**
Sugar content = Percentage × Quantity
= (25/100) × 120
= (1/4) × 120
= 30g sugar
**Method 3: Using Ratios**
25% sugar means 25g sugar per 100g biscuits, which means:
**Visual Representation (Bar Model):**
```
0 25% 50% 75% 100%
|--------|--------|--------|--------|
0 30g 60g 90g 120g (Sugar)
(120g biscuits total)
```
For **Madhav (95g with 35% sugar):**
**Method 1: Find sugar content per gram**
100g of biscuits has 35g sugar
1g of biscuits has 35/100 g sugar
95g of biscuits has (35/100) × 95 = 33.25g sugar
**Method 2: Using Proportion**
35/100 = s/95
s = (35/100) × 95 = 33.25g sugar
**Conclusion:** Madhav ate more sugar (33.25g > 30g), even though his biscuits had a higher percentage, because he ate less total quantity.
**Formula:** y% of some value x is given by:
**y% of x = (y/100) × x**
**Examples:**
#### Strategy 1: Recognizing Percentages as Simple Fractions
**Key Equivalences:**
**Advantage:** Once you recognize a percentage as a simple fraction, finding the percentage becomes easy.
**Example: Finding 25% of 40**
Since 25% = 1/4:
25% of 40 = (1/4) × 40 = 10
**Example: Finding 50% of 80**
Since 50% = 1/2:
50% of 80 = (1/2) × 80 = 40
#### Strategy 2: Using Doubling and Halving
**Key Relationship:** 20% is double 10%. This always holds because:
20% = 20/100 and 10% = 10/100
And 20/100 = 2 × (10/100)
**Practice Table for Mental Computation:**
For finding different percentages of values, use this table as reference:
| Value | 25% | 10% | 20% | 5% |
|-------|-----|-----|-----|-----|
| 100 | 25 | 10 | 20 | 5 |
| 200 | 50 | 20 | 40 | 10 |
| 50 | 12.5 | 5 | 10 | 2.5 |
| 80 | 20 | 8 | 16 | 4 |
| 10 | 2.5 | 1 | 2 | 0.5 |
| 35 | 8.75 | 3.5 | 7 | 1.75 |
| 287 | 71.75 | 28.7 | 57.4 | 14.35 |
**Interesting Observation:** Notice that 20% of a value is exactly double 10% of the same value. For example:
This relationship holds **always** because doubling the percentage (20% vs 10%) doubles the result.
#### Strategy 3: Using Additive Properties
**Key Property:** When percentages are additive, the amounts are also additive.
**(20% of y) + (5% of y) = 25% of y**
**Mathematical Proof:**
(20/100 × y) + (5/100 × y) = (25/100 × y)
(20 + 5)/100 × y = 25/100 × y ✓
**Practical Use for Mental Math:**
To find 15% of a value:
15% = 10% + 5%
So: 15% of 80 = 10% of 80 + 5% of 80 = 8 + 4 = 12
To find 75% of a value:
75% = 50% + 25%
So: 75% of 120 = 50% of 120 + 25% of 120 = 60 + 30 = 90
To find 40% of a value:
40% = 20% + 20% (or 40% = 50% - 10%)
So: 40% of 100 = 20 + 20 = 40 (or 50 - 10 = 40)
To find 90% of a value:
90% = 100% - 10%
So: 90% of 50 = 50 - 5 = 45
To find 70% of a value:
70% = 50% + 20%
So: 70% of 200 = 100 + 40 = 140
**Important Connection:** Fractions, Decimals, and Percentages (FDP) are three different ways of representing the same quantity.
**Example: Three Representations of One-Half**
**Why They're Equivalent:**
1/2 = 0.5 = 50/100 = 50%
#### Converting Between Percentages and Decimals
**Relationship:** Multiply/divide by 100 to convert between percentages and decimals.
**To Convert Percentage to Decimal:** Divide by 100
**To Convert Decimal to Percentage:** Multiply by 100
#### Complete FDP Conversion Table
| Percentage | Fraction | Decimal |
|------------|----------|---------|
| 50% | 50/100 = 1/2 | 0.5 |
| 100% | 100/100 = 1 | 1.0 |
| 25% | 25/100 = 1/4 | 0.25 |
| 75% | 75/100 = 3/4 | 0.75 |
| 10% | 10/100 = 1/10 | 0.1 |
| 1% | 1/100 | 0.01 |
| 5% | 5/100 = 1/20 | 0.05 |
| 43% | 43/100 | 0.43 |
| 20% | 20/100 = 1/5 | 0.2 |
| 33.33% | 33.33/100 ≈ 1/3 | 0.3333... |
**Using Decimals to Find Percentages:** To find 50% of 24:
This is exactly the same as (1/2) × 24 = 12.
#### Multiple Methods to Find Percentage of a Quantity
**Example 3: Finding 80% of 75 marks**
Real-life context: A test has maximum marks of 75. Students need 80% or above for an A grade. How much should Zubin score at least?
**Method 1: Using Fraction Multiplication**
80% of 75 = (80/100) × 75
= (4/5) × 75
= (4 × 75)/5
= 300/5
= 60
**Method 2: Using Decimal Multiplication**
80% of 75 = 0.8 × 75
= 60
**Method 3: Using Proportional Reasoning**
If out of 100 marks, the minimum is 80 marks:
Out of 75 marks, the minimum is:
= 75 × (80/100)
= 75 × 0.8
= 60
**Answer:** Zubin should score at least 60 marks to get an A grade.
**All three methods give the same answer.** Choose whichever method feels most natural to you.
**Converting Ratios to Percentages**
When quantities are given as a ratio, we can convert them to percentages by:
1. Converting the ratio to fractions
2. Finding each part's percentage of the whole
**Example 4: Millet Kanji Recipe**
Real-life context: To prepare millet kanji (porridge), the ratio of millet : water is 2:7.
**Question 1:** What percentage does millet constitute?
**Step 1: Find the total parts**
Millet : Water = 2 : 7
Total parts = 2 + 7 = 9
**Step 2: Express each component as a fraction of total**
**Step 3: Convert to percentages**
Millet percentage = (2/9) × 100 = 22.22%
Water percentage = (7/9) × 100 = 77.78%
**Verification:** 22.22% + 77.78% = 100% ✓
**Question 2:** If 500 ml of mixture is to be made, how much millet should be used?
**Step 1: Recall millet is 22.22% of the mixture**
**Step 2: Find 22.22% of 500 ml**
Amount of millet = 22.22% of 500
= (22.22/100) × 500
= (2/9) × 500
= 1000/9
= 111.11 ml
**Estimation Strategy:** Before calculating, estimate:
**Why Estimate First?** Estimation helps develop number sense and can prevent silly mistakes.
**Estimation Strategy:** Before calculating 25% of 160:
**Example with Bar Models:**
For 80% of 75:
**Example 5: Journey Problem**
Real-life context: A cyclist cycles from Delhi to Agra and completes 40% of the journey, covering 92 km.
**Question:** How many more kilometers does the cyclist need to travel?
**Step 1: Draw the Bar Model**
```
Delhi ├─────40%─────┤├─────60%─────┤ Agra
└─────92 km────┘└──────?──────┘
└────────────Total Distance─────────────┘
(100%)
```
**Step 2: Estimate Before Solving**
**Step 3: Solve Using Different Methods**
**Method 1: Finding the Whole, Then the Remaining Part**
If 40% = 92 km, find 100%:
(40/100) = (92/d) [where d is total distance]
d = 92 × (100/40)
d = 92 × 2.5
d = 230 km
Remaining distance = 230 - 92 = 138 km
**Method 2: Using Doubling Strategy**
40% is 92 km
20% is 92 ÷ 2 = 46 km
60% is 92 + 46 = 138 km (using additive property)
**Method 3: Using Proportion**
If 40% corresponds to 92 km:
60% corresponds to x km
40/60 = 92/x
40x = 60 × 92
x = (60 × 92)/40
x = 138 km
**Method 4: Using Variables**
Let x = remaining distance
Total distance = x + 92
Since 92 is 40% of total:
(40/100) × (x + 92) = 92
(x + 92) = 92 × (100/40)
x + 92 = 230
x = 138 km
**Answer:** The cyclist needs to travel 138 more kilometers.
---
**Key Insight:** A percentage can be greater than 100%. This happens when the compared quantity is **larger than the base quantity (reference)**.
**What It Means:** If something is 120% of a value, it means it is 20% more than that value.
#### Real-Life Example: Shop Sales Performance
**Example 6: Analyzing Daily Sales Against Target**
Real-life context: Kishanlal's garment shop has a daily sales target of ₹5000.
**Day 1 Sales: ₹2000**
Percentage achieved = (Sales/Target) × 100
= (2000/5000) × 100
= (2/5) × 100
= 40%
**Interpretation:** On Day 1, sales were only 40% of the target. Alternatively, the shop was 60% short of its target.
**Bar Model:**
```
0% 100%
├────────────────────────────┤
Target: ₹5000
├─40%──┤ (₹2000)
├──────────60% short──────────┤
```
**Day 2 Sales: ₹3500**
Percentage achieved = (3500/5000) × 100
= (7/10) × 100
= 70%
**Interpretation:** On Day 2, sales were 70% of the target. The shop was 30% short.
**Day 3 Sales: ₹5000**
Percentage achieved = (5000/5000) × 100
= 1 × 100
= 100%
**Interpretation:** The shop exactly met its target on Day 3.
**Bar Model:**
```
0% 100%
├────────────────────────────┤
Target: ₹5000
├─────────100%─────────┤
Sales: ₹5000 (exactly matches target)
```
**Day 4 Sales: ₹6000**
Percentage achieved = (6000/5000) × 100
= (6/5) × 100
= 1.2 × 100
= 120%
**Interpretation:** On Day 4, sales were 120% of the target. The shop exceeded its target by 20%.
**Analysis:**
**Bar Model:**
```
0% 100% 120%
├──────────────────────────────────┤
Target: ₹5000
├──────────────5000──────────┤├20%-┤
Sales: ₹6000 (100% + 20% more)
```
**Day 5 Sales: ₹7800**
Percentage achieved = (7800/5000) × 100
= (78/50) × 100
= 1.56 × 100
= 156%
**Interpretation:** Sales were 156% of target, meaning 56% above target.
**Day 6 Sales: ₹9550**
Percentage achieved = (9550/5000) × 100
= (955/500) × 100
= 1.91 × 100
= 191%
**Interpretation:** Sales were 191% of target, meaning 91% above target.
**Example:** If the shop achieved 150% of its target on Day 7, what were the sales?
150% of ₹5000 = (150/100) × 5000
= 1.5 × 5000
= ₹7500
**Interpretation:** 150% means 1 complete target (100%) plus another half (50%), so it's 1.5 times the target.
**Example:** If the shop achieved 210% of its target on Day 8, what were the sales?
210% of ₹5000 = (210/100) × 5000
= 2.1 × 5000
= ₹10,500
**Interpretation:** 210% means more than double the target (which would be 200%), specifically 2.1 times the target.
**Complete Table for Various Percentages:**
| Percentage | Fraction | Decimal |
|------------|----------|---------|
| 90% | 90/100 = 9/10 | 0.9 |
| 100% | 100/100 = 1 | 1.0 |
| 110% | 110/100 = 11/10 | 1.1 |
| 150% | 150/100 = 3/2 | 1.5 |
| 200% | 200/100 = 2 | 2.0 |
| 173% | 173/100 | 1.73 |
| 250% | 250/100 = 5/2 | 2.5 |
| 305% | 305/100 | 3.05 |
| 358% | 358/100 | 3.58 |
| 28.9% | 28.9/100 | 0.289 |
**Key Observations:**
**Alternative Ways to Express Percentages Greater than 100%:**
Instead of "120% of target," you can say:
Instead of "200% of value," you can say:
**Example 7: Harvest Comparison**
Real-life context: A farmer's wheat harvest comparison.
**Given:**
**Question:** What percentage of last year's harvest is this year's harvest?
**Solution:**
Percentage = (This year's harvest / Last year's harvest) × 100
= (650/260) × 100
= (65/26) × 100
= 2.5 × 100
= 250%
**Interpretation:** This year's harvest is 250% of last year's harvest.
This means:
**Verification:** 2.5 × 260 = 650 ✓
---
**y% of x = (y/100) × x**
**Fraction a/b = (a/b) × 100%**
**Percentage z% = z/100** (can be simplified)
**Percentage = (Amount/Total) × 100**
**Total = Amount ÷ (Percentage/100)**
**Amount = (Percentage/100) × Total**
**Fraction = Decimal = Percentage/100**
**If percentage > 100%, then the quantity is larger than the base quantity.**
**Percentage - 100% = amount above base (as percentage)**
---
**❌ Wrong:** Comparing "Madhu ate 25% sugar" with "Madhav ate 35% sugar" without knowing quantities eaten.
**✓ Correct:** Always identify the base quantity (the whole that the percentage refers to).
**❌ Wrong:** If 40% is 92 km, finding total as 40/92 × 100 = 43.5
**✓ Correct:** Total = 92 ÷ (40/100) = 92 × (100/40) = 230 km
**
Q1. What does 35% mean?
Answer: A — Percent means 'per hundred', so 35% = 35/100, which means 35 out of every 100.
Q2. Which of the following is equivalent to 50%?
Answer: B — 50% = 50/100 = 1/2, so 50% is the same as the fraction one-half.
Q3. Express 3/5 as a percentage.
Answer: C — 3/5 = (3 × 20)/(5 × 20) = 60/100 = 60%, or multiply: (3/5) × 100 = 60.
Q4. What is 25% of 80?
Answer: B — 25% of 80 = (25/100) × 80 = (1/4) × 80 = 20.
Q5. Nandini has 25 marbles, of which 15 are white. What percentage of her marbles are white?
Answer: C — Percentage = (15/25) × 100 = (3/5) × 100 = 60%.
Q6. In a school, 15 of the 80 students come by walking. What percentage of students walk to school?
Answer: B — Percentage = (15/80) × 100 = (3/16) × 100 = 18.75%.
Q7. Madhav ate 95 g of biscuits with 35% sugar. How much sugar did he consume?
Answer: B — 35% of 95 g = (35/100) × 95 = 33.25 g.
Q8. A test has maximum marks 75. If a student needs 80% to pass, how many marks must they score?
Answer: C — 80% of 75 = (80/100) × 75 = (4/5) × 75 = 60 marks.
Q9. Which statement correctly compares 50% and 5%?
Answer: C — 50% (50 out of 100) is greater than 5% (5 out of 100) as percentages are proportions independent of the base.
Q10. Madhu ate 120 g of biscuits with 25% sugar and Madhav ate 95 g with 35% sugar. Who consumed more sugar?
Answer: B — Madhu consumed 25% of 120 g = 30 g sugar; Madhav consumed 35% of 95 g = 33.25 g sugar; therefore Madhav consumed more.
What does the symbol '%' mean and where does the word come from?
The symbol '%' means 'per cent' which comes from Latin 'per centum' meaning 'out of hundred' or 'by the hundred'.
How do you convert the fraction 3/4 into a percentage?
Multiply 3/4 by 100 to get (3/4) × 100 = 75%, or find the equivalent fraction with denominator 100: (3 × 25)/(4 × 25) = 75/100 = 75%.
Write the general formula to find y% of some value z.
y% of z = (y/100) × z.
What is the relationship between 25%, 1/4, and 0.25?
They are all equivalent: 25% = 25/100 = 1/4 = 0.25.
Why is 100 chosen as the denominator for percentages instead of 10 or 1000?
100 is the sweet spot because our number system is base 10, it connects easily to decimals, it is large enough to give detail, yet simple enough to understand mentally.
If Madhu ate 120 g of biscuits with 25% sugar, how much sugar did she eat?
25% of 120 g = (25/100) × 120 = 30 g of sugar.
When comparing two percentages of different wholes, what must you do first?
You must find the actual quantities (not just the percentages) because percentage is a proportion and depends on the total amount.
Express 24% as a fraction in simplest form.
24% = 24/100 = 6/25.
If 80% of a test has maximum marks 75, what is the minimum score for grade A?
80% of 75 = (80/100) × 75 = 60 marks.
What relationship always holds between 20%, 5%, and 25% of any value y?
(20% of y) + (5% of y) = 25% of y, because (20/100 + 5/100) × y = (25/100) × y.
What does 45% mean? [1 mark]
Per cent means 'out of hundred'. Write as a fraction with denominator 100.
Convert 7/20 into a percentage. Show your method. [2 marks]
Find equivalent fraction with denominator 100 by multiplying numerator and denominator by 5, or multiply the fraction by 100.
In a school library, there are 240 books. If 30% of the books are fiction, how many fiction books are there? Show your working step by step. [3 marks]
Use formula: 30% of 240 = (30/100) × 240. Simplify the fraction first to make calculation easier, or convert to decimal 0.3 × 240.
A shopkeeper gives a 20% discount on a shirt that costs ₹500. How much does the customer pay? Explain the concept of percentage discount and show all your steps. Also state why percentages are more useful than fractions for comparing such quantities. [5 marks]
First find 20% of 500 using (20/100) × 500. Subtract from original price. Explain that percentage makes comparison standard (out of 100) regardless of original value, unlike fractions which vary with different denominators. Also mention that the number system base 10 makes 100 convenient for understanding.
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