Proportion word problems with full solutions:

DOWNLOAD HERE - PDF FILE

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1. A pack of 4 pens costs $6. How much would 10 pens cost?

Solution:
Cost per pen = $6 ÷ 4 = $1.50
Cost for 10 pens = 10 × $1.50 = $15
✅ Answer: c) $15

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2. A bakery sells 3 cupcakes for $9. How much for 7 cupcakes?

Solution:
Cost per cupcake = $9 ÷ 3 = $3
Cost for 7 cupcakes = 7 × $3 = $21
✅ Answer: b) $21

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3. 6 notebooks cost $18. How much would 2 notebooks cost?

Solution:
Cost per notebook = $18 ÷ 6 = $3
Cost for 2 notebooks = 2 × $3 = $6
✅ Answer: c) $6

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4. A store offers 10 pencils for $5. What would 25 pencils cost?

Solution:
Cost per pencil = $5 ÷ 10 = $0.50
Cost for 25 pencils = 25 × $0.50 = $12.50
✅ Answer: b) $12.50

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5. 8 chocolate bars cost $16. How much do 5 chocolate bars cost?

Solution:
Cost per bar = $16 ÷ 8 = $2
Cost for 5 bars = 5 × $2 = $10
✅ Answer: c) $10

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6. A bundle of 12 apples costs $24. What’s the cost of 6 apples?

Solution:
Cost per apple = $24 ÷ 12 = $2
Cost for 6 apples = 6 × $2 = $12
✅ Answer: b) $12

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7. 7 liters of juice cost $14. What is the price for 3 liters?

Solution:
Cost per liter = $14 ÷ 7 = $2
Cost for 3 liters = 3 × $2 = $6
✅ Answer: b) $6

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8. A box of 9 markers costs $27. How much would 4 markers cost?

Solution:
Cost per marker = $27 ÷ 9 = $3
Cost for 4 markers = 4 × $3 = $12
✅ Answer: c) $12

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9. You get 2 movie tickets for $18. How much would 5 tickets cost?

Solution:
Cost per ticket = $18 ÷ 2 = $9
Cost for 5 tickets = 5 × $9 = $45
✅ Answer: b) $45

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10. A grocery store sells 6 cans of soup for $9. What is the cost of 10 cans?

Solution:
Cost per can = $9 ÷ 6 = $1.50
Cost for 10 cans = 10 × $1.50 = $15
✅ Answer: c) $15

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proportion word problems with full solutions 

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1. A recipe calls for 3 cups of flour to make 12 cookies.

How many cups are needed for 36 cookies?

Solution:
Set up proportion:
$\frac{3 \text{ cups}}{12 \text{ cookies}} = \frac{x \text{ cups}}{36 \text{ cookies}}$
Cross-multiply:
$12x = 3 × 36 = 108$
$x = \frac{108}{12} = 9$
✅ Answer: 9 cups

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2. A smoothie recipe uses 2 bananas for 4 servings.

How many bananas are needed for 10 servings?

Solution:
$\frac{2}{4} = \frac{x}{10} \Rightarrow 4x = 20 \Rightarrow x = 5$
✅ Answer: 5 bananas

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3. It takes 5 cups of rice to serve 8 people.

How many cups are needed to serve 20 people?

Solution:
$\frac{5}{8} = \frac{x}{20} \Rightarrow 8x = 100 \Rightarrow x = 12.5$
✅ Answer: 12.5 cups

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4. A recipe makes 6 muffins using 2 eggs.

How many eggs are needed for 18 muffins?

Solution:
$\frac{2}{6} = \frac{x}{18} \Rightarrow 6x = 36 \Rightarrow x = 6$
✅ Answer: 6 eggs

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5. 4 tablespoons of sugar make 8 cups of lemonade.

How many tablespoons are needed for 20 cups?

Solution:
$\frac{4}{8} = \frac{x}{20} \Rightarrow 8x = 80 \Rightarrow x = 10$
✅ Answer: 10 tablespoons

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6. A cake recipe uses 1.5 cups of milk for 6 servings.

How much milk is needed for 18 servings?

Solution:
$\frac{1.5}{6} = \frac{x}{18} \Rightarrow 6x = 27 \Rightarrow x = 4.5$
✅ Answer: 4.5 cups

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7. A soup recipe needs 2.5 liters of water for 5 bowls.

How much for 8 bowls?

Solution:
$\frac{2.5}{5} = \frac{x}{8} \Rightarrow 5x = 20 \Rightarrow x = 4$
✅ Answer: 4 liters

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8. 6 scoops of ice cream serve 3 people.

How many scoops for 9 people?

Solution:
$\frac{6}{3} = \frac{x}{9} \Rightarrow 3x = 54 \Rightarrow x = 18$
✅ Answer: 18 scoops

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9. A batch of dough uses 4 cups of flour to make 24 rolls.

How much flour is needed for 60 rolls?

Solution:
$\frac{4}{24} = \frac{x}{60} \Rightarrow 24x = 240 \Rightarrow x = 10$
✅ Answer: 10 cups

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10. 5 liters of paint covers 15 square meters.

How much paint is needed for 45 square meters?

Solution:
$\frac{5}{15} = \frac{x}{45} \Rightarrow 15x = 225 \Rightarrow x = 15$
✅ Answer: 15 liters

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Simple interest quiz with a twist

Each question comes with multiple choices—try to answer first, then check the solution after!

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1. Emily borrowed $2,000 at a rate of 6% for 4 years. What is the total interest?

A. $360
B. $480
C. $540
D. $600

Formula:

$$\text{Interest} = 2000 \times 0.06 \times 4 = \boxed{480}$$

✅ Correct Answer: B. $480

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2. Jack invested $1,500 for 2 years and earned $180 in interest. What was the rate?

A. 5%
B. 6%
C. 8%
D. 12%

$$\text{Rate} = \frac{180}{1500 \times 2} = 0.06 = \boxed{6\%}$$

✅ Correct Answer: B. 6%

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3. A loan of $800 earns $96 in interest at 4% interest. How long was the loan?

A. 2 years
B. 3 years
C. 4 years
D. 5 years

$$\text{Time} = \frac{96}{800 \times 0.04} = \frac{96}{32} = \boxed{3}$$

✅ Correct Answer: B. 3 years

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4. Sarah paid $525 in interest on a 5-year loan at 7%. What was the original principal?

A. $1,200
B. $1,400
C. $1,500
D. $1,700

$$\text{Principal} = \frac{525}{0.07 \times 5} = \frac{525}{0.35} = \boxed{1500}$$

✅ Correct Answer: C. $1,500

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5. Tom earned $600 in interest over 3 years at 10%. How much did he invest?

A. $1,800
B. $2,000
C. $2,400
D. $2,800

$$\text{Principal} = \frac{600}{0.10 \times 3} = \frac{600}{0.30} = \boxed{2000}$$

✅ Correct Answer: B. $2,000

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Class 8 NCERT bridge course Answers Activity W 5.3 Logic Clue Hunt on the Hundred Square

 Activity W 5.3 -  Logic Clue Hunt on the Hundred Square 

For this activity, students work in pairs or small groups. 

The students may draw a hundred square as shown below:

Procedure 

 The following clues may be written on the blackboard: 

 The number is greater than 9. 

 The number is not a multiple of 10. 

 The number is a multiple of 8. ¾ The number is even. 

The number is not a multiple of 11. 

 The number is less than 175. 

 Its ones digit is larger than its tens digit. 

 Its tens digit is odd.

Part A 

 Tell the students 

 How have a number in your mind that is on the hundred squares but you are not going to tell them what it is. 

They have to ask you for any four clues out of the given eight clues. 

With every clue they speak out, you will say just ‘YES’ or ‘NO’. 

 Try to find the set of four clues that help them to find the number in your mind. 

 Give a chance to each group to do this. 

Strategy Tip: Encourage teams to choose clues that narrow the number range quickly.

Part B 

Four of the given clues are true but they do not help in finding the number.

 Find those numbers. 

Reflection 

Consider the questions that led the students being interested and able to progress, and those you needed to clarify. 

Such reflection always helps you engage the students to find mathematics interesting and enjoyable. 

If they do not understand and do something, they are less likely to become involved.

Part A: Which clues help identify the number?

We are given 8 clues. The goal is to identify one specific number using only 4 well-chosen clues. Here’s how you can think through the process:

Let's analyze each clue for usefulness:

Clue Analysis:

1. The number is greater than 9
   ➤ Too broad — eliminates only numbers 1–9.
   Not very useful.

2. The number is not a multiple of 10
   ➤ Removes numbers ending in 0 (e.g., 10, 20, ..., 200).
   Somewhat useful.

3. The number is a multiple of 8
   ➤ Strong clue. Narrows down to numbers like 8, 16, 24, 32, etc.
   Very useful!

4. The number is even
   ➤ All multiples of 8 are even already.
   Redundant if Clue 3 is chosen.

5. The number is not a multiple of 11
   ➤ Excludes numbers like 11, 22, 33, ..., 198.
   Somewhat useful.

6. The number is less than 175
   ➤ Trims the upper end.
   Useful for narrowing down.

7. Its ones digit is larger than its tens digit
   ➤ Powerful filter (e.g., 13, 24, 57, but not 31, 43).
   Very useful!

8. Its tens digit is odd
   ➤ Limits numbers to those with tens digit as 1, 3, 5, 7, or 9.
   Very useful!

 Example: Find the Hidden Number

Let's pick a number that satisfies the following 4 helpful clues:

• Clue 3: Multiple of 8

• Clue 6: Less than 175

• Clue 7: Ones digit > Tens digit

• Clue 8: Tens digit is odd

Let’s test numbers that are:

• Multiples of 8

• Less than 175

• Have ones digit > tens digit

• Tens digit is odd

Example: 136

• Multiple of 8 

• Less than 175 

• Ones digit (6) > Tens digit (3) 

• Tens digit (3) is odd 
  → 136 is a valid hidden number!

Part B: Which clues are always true but not useful?

These clues may be true for many numbers, but don’t help narrow the list:

1. Clue 1: Greater than 9 → Always true for almost all 2- or 3-digit numbers.

2. Clue 2: Not a multiple of 10 → Excludes just a few (10, 20, ..., 200).

3. Clue 4: Even → Already covered by “multiple of 8.”

4. Clue 5: Not a multiple of 11 → Useful only if the number was close to a multiple of 11.

So, the 4 clues that don’t help much, even if true, are:

• Clue 1 (Greater than 9)

• Clue 2 (Not a multiple of 10)

• Clue 4 (Even)

• Clue 5 (Not a multiple of 11)

These clues are logically true for many numbers but don’t help you zero in on the correct number efficiently.

Class 8 NCERT bridge course Answers Activity W 5.2 Number sense

 Activity W 5.2  Number sense

Number sense involves giving meaning to numbers, that is, knowing about how they relate to each other and their relative magnitudes. 

 Having a sense of number is vital for the understanding of numerical aspects of the world. 

Here are some ideas to develop and strengthen students’ sense of numbers. 

Activity 1: Two-Digit Number Trick

Objective: Discover a pattern when reversing and adding two-digit numbers.

Procedure 

  Ask the students to choose a two-digit number. 

 Tell them to reverse the digits to get a new number. 

 Add this new number to the original number. 

 Ask students to check for their divisibility by a number.

 Check if every student gets a number by which the sum obtained in step 3 is divisible. 

 Discuss why this happens! 

Steps:

1. Ask students to choose any two-digit number (e.g., 52).

2. Reverse the digits to form a new number (e.g., 25).

3. Add the original and reversed number (e.g., 52 + 25 = 77).

4. Ask students to check the divisibility of the result.

Challenge Question:

• Is the result always divisible by a specific number?

Answer: yes, The sum is always divisible by 11.

• (Hint: Try with different numbers – 34 + 43 = 77, 61 + 16 = 77...)

✨ Discover: The sum is always divisible by 11. Why does this happen?

Why does this happen?

Answer: 

Let the two-digit number be 10a + b, where a is the tens digit and b is the units digit.

The reverse of the number is 10b + a.

Now add the two:  (10a+b)+(10b+a)=11a+11b=11(a+b)

This sum is clearly divisible by 11, because 11 is a common factor.

Activity 2: Three-Digit Number Difference

Objective: Explore divisibility through subtraction of reversed numbers.

Procedure 

 Ask students to think of a three-digit number. 

 Now, they should make a new number by putting the digits in the reverse order. 

 Subtract the smaller number from the larger one. 

 Ask the students to check by which number the difference so obtained is divisible. Which other multiple of this divisor will divide the difference? 

 Discuss how this happens! 

Steps:

1. Choose any three-digit number (e.g., 741).

2. Reverse the digits (e.g., 147).

3. Subtract the smaller from the larger (e.g., 741 - 147 = 594).

4. Ask: Which number divides this difference?

Question: What is the difference divisible by?

Answer: The difference is always divisible by 99.

Follow-Up:

• Try multiple three-digit numbers. What do you notice?

• Is there a common divisor?

✨ Discover: The difference is always divisible by 99 (or sometimes 9 and 11).

Why does it work?

Let the number be 100a + 10b + c, and the reverse is 100c + 10b + a.

Now subtract the smaller from the larger:

$$(100a+10b+c)-(100c+10b+a)=99a-99c=99(a-c)$$

So the difference is always divisible by 99.

🧠 Bonus: Since 99 = 9 × 11, it's also divisible by 9 and 11.

Activity 3: Rotating Digits of a Three-Digit Number

Objective: Explore rotational patterns in digits and their divisibility.

Procedure 

 Students may be asked to think of any 3-digit number (abc). 

 Now, using this number, students may be asked to form two more 3-digit numbers (cab, bca).

Now, add the three numbers so formed. 

 Students may explore the smallest number by which it will be divisible. 

 Discuss how this happens!

Steps:

1. Think of a three-digit number (e.g., 231).

2. Create two more numbers by rotating the digits:

   • (cab → 312)

   • (bca → 123)

3. Add the three numbers:

   • 231 + 312 + 123 = 666

4. Ask: What is the smallest number that always divides the sum?

Answer: The sum is always divisible by 37 (and also 3 and 9).

✨ Discover: The sum is divisible by 37 (and also 3 and 9)!

Why does it work?

Let the three-digit number be abc:

• Its numerical value is 100a + 10b + c

• Rotation 1 (cab): 100c + 10a + b

• Rotation 2 (bca): 100b + 10c + a

Now add all three:

$$(100a+10b+c)+(100c+10a+b)+(100b+10c+a)=111a+111b+111c=111(a+b+c)$$ $$\text{<br>}\text{<br>}$$