Word Problems On Agez

 

Mastering Age Word Problems: Concepts, Hints, and Examples

Age problems are a classic staple of algebra. They might look like riddles at first, but once you break them down into variables and equations, they become simple puzzles to solve.

Introduction to the Concept of Ages

At its core, an age problem is just a linear equation in disguise. The fundamental concept relies on a single, unchangeable truth: time moves at the same rate for everyone.

If you are 5 years older than your sister today, you will still be 5 years older than her 20 years from now. When solving these problems, we generally deal with three distinct time frames:

• The Past: indicated by phrases like "5 years ago" or "in 2018".
• The Present: indicated by phrases like "currently", "is", or "now".
• The Future: indicated by phrases like "in 10 years", "hence", or "will be".

Key Hints for Solving Age Calculations

To translate a word problem into a workable mathematical equation, keep these handy rules in mind:

• Define the Present First: Always let the present age of a person be your primary variable (e.g., J for John). It makes navigating backwards or forwards in time much easier.
• "Years Ago" means Subtraction: If a person's current age is x, their age n years ago was x - n.
• "Years Hence" means Addition: If a person's current age is x, their age n years from now will be x + n.
• Apply Time Changes to Everyone: If a problem moves 5 years into the future, you must add 5 to every single person's age.
• Watch the Verbs: "Is/Was/Will be" translate to equals (=), and "Times" translates to multiplication (×).

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5 Simple Practice Examples

Part 1: Age of One Person (Past vs. Future)

Example 1: Finding Present Age from the Future

Problem: In 12 years, John will be three times as old as he is right now. How old is John today?

Setup: Let John's current age be J. In 12 years, his age will be J + 12.

Equation: J + 12 = 3J

Calculation:
12 = 2J
J = 6

Answer: John is 6 years old today.

Example 2: Finding Present Age from the Past

Problem: Five years ago, Sarah was half the age she will be next year. How old is Sarah now?

Setup: Let Sarah's current age be S. Five years ago she was S - 5. Next year she will be S + 1.

Equation: S - 5 = 0.5(S + 1)

Calculation:
2(S - 5) = S + 1
2S - 10 = S + 1
S = 11

Answer: Sarah is 11 years old now.

Part 2: Relationship Between Three People

Example 3: Sum of Ages with Ratios

Problem: The total age of Amy, Brad, and Charlie is 45 years. Brad is twice as old as Amy, and Charlie is 5 years older than Brad. How old is Amy?

Setup: Let Amy's age be A. Therefore, Brad is 2A, and Charlie is 2A + 5.

Equation: A + 2A + (2A + 5) = 45

Calculation:
5A + 5 = 45
5A = 40
A = 8

Answer: Amy is 8 years old.

Example 4: Moving into the Past with Three People

Problem: Tom is 10 years older than Jerry, and Jerry is 4 years older than Spike. Three years ago, the sum of their ages was 30. How old is Spike now?

Setup: Let Spike's age be S. Jerry is S + 4. Tom is S + 14.

Equation (Three years ago): (S - 3) + (S + 4 - 3) + (S + 14 - 3) = 30

Calculation:
(S - 3) + (S + 1) + (S + 11) = 30
3S + 9 = 30
3S = 21
S = 7

Answer: Spike is 7 years old.

Example 5: Future Multipliers

Problem: Liam is twice as old as Noah, and Noah is twice as old as Mason. In 2 years, the sum of Liam and Mason's ages will equal 3 times Noah's current age. How old is Noah?

Setup: Let Mason = M. Noah = 2M. Liam = 4M.

Equation: (4M + 2) + (M + 2) = 3(2M)

Calculation:
5M + 4 = 6M
M = 4
Noah's age = 2(4) = 8

Answer: Noah is 8 years old.

Problems On Profit And Loss

 

Profit and Loss: The Basics

• Profit: Occurs when a business sells a product or service for more than it cost to buy. It represents financial gain.
• Loss: Occurs when a business sells a product or service for less than its cost price. It represents financial deficit.

The Key Formulas

• Profit = Selling Price (SP) - Cost Price (CP)
• Loss = Cost Price (CP) - Selling Price (SP)
• Profit Percentage (%) = (Profit / CP) × 100
• Loss Percentage (%) = (Loss / CP) × 100

8 Simple Practice Questions & Solutions

Q1. Simple Profit

A shopkeeper buys a book for $200 and sells it for $250. Find the profit.

Solution:
CP = $200, SP = $250
Profit = SP - CP = 250 - 200 = 50
Answer: The profit is $50.

Q2. Simple Loss

A cycle bought for $1,200 is sold for $1,000. Find the loss.

Solution:
CP = $1,200, SP = $1,000
Loss = CP - SP = 1200 - 1000 = 200
Answer: The loss is $200.

Q3. Profit Percentage

An item bought for $80 is sold for $100. Find the profit percentage.

Solution:
CP = $80, SP = $100
Profit = 100 - 80 = 20
Profit % = (20 / 80) × 100 = 25%
Answer: The profit percentage is 25%.

Q4. Loss Percentage

A toy bought for $50 is sold for $40. Find the loss percentage.

Solution:
CP = $50, SP = $40
Loss = 50 - 40 = 10
Loss % = (10 / 50) × 100 = 20%
Answer: The loss percentage is 20%.

Q5. Finding Selling Price (Profit)

A man buys a watch for $500 and wants to make a 10% profit. What should the selling price be?

Solution:
CP = $500, Profit % = 10%
Profit Amount = 10% of 500 = $50
SP = CP + Profit = 500 + 50 = 550
Answer: The selling price should be $550.

Q6. Finding Selling Price (Loss)

A table bought for $300 is sold at a loss of 15%. Find the selling price.

Solution:
CP = $300, Loss % = 15%
Loss Amount = 15% of 300 = $45
SP = CP - Loss = 300 - 45 = 255
Answer: The selling price is $255.

Q7. Finding Cost Price (Profit)

An article is sold for $120 at a profit of 20%. Find its cost price.

Solution:
SP = $120, Profit % = 20%
SP = CP × 1.20 → 120 = CP × 1.20
CP = 120 / 1.20 = 100
Answer: The cost price is $100.

Q8. Finding Cost Price (Loss)

By selling a bag for $90, a shopkeeper incurs a loss of 10%. Find the cost price.

Solution:
SP = $90, Loss % = 10%
SP = CP × 0.90 → 90 = CP × 0.90
CP = 90 / 0.90 = 100
Answer: The cost price is $100.

Solving Problems On Percentages

 

🔢 Understanding Percentages

A percentage is a number or ratio expressed as a fraction of 100. It is derived from the Latin phrase per centum, meaning "by the hundred," and is denoted using the percent sign (%).

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💡 Practice Questions & Solutions

Q1: What is 20% of 80?

Solution

Convert 20% to a decimal (0.20) and multiply by 80:
0.20 × 80 = 16

Q2: Convert 3/5 to a percentage.

Solution

Divide 3 by 5, then multiply the decimal by 100:
3 ÷ 5 = 0.60
0.60 × 100 = 60%

Q3: A basket has 15 red apples and 5 green apples. What percentage are green?

Solution

Total apples = 20. Divide green apples by the total:
5 ÷ 20 = 0.25
0.25 × 100 = 25%

Q4: A jacket costs $100 and is 15% off. How much do you save?

Solution

Find 15% of 100:
0.15 × 100 = $15

Q5: A population grows from 500 to 550. What is the percentage increase?

Solution

Increase = 50. Divide by the original population size:
50 ÷ 500 = 0.10
0.10 × 100 = 10%

Solving Ratio Propblems

 

Introduction to Ratios for Primary School

A ratio is a way to compare two or more quantities to show how much of one thing there is compared to another. It tells us the relationship between the sizes of two groups.

Imagine you are serving food at a party. For every 3 plates of Jollof rice, you serve 2 plates of Fried rice. The ratio of Jollof rice to Fried rice is 3 to 2.

How We Write Ratios

We can write a ratio in three different ways:

• With the word "to" → 3 to 2
• With a colon symbol → 3 : 2
• As a fraction → ³⁄₂

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The Ratio Formula

Ratio = (Quantity of A) ÷ (Quantity of B)

Important Rule: Ratios should always be simplified to their lowest terms by dividing both sides by the biggest number that can go into them perfectly.

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Practice Questions

Question 1 (Easy • Fruits): In a market basket at Balogun market, there are 12 mangos and 18 oranges. What is the ratio of mangos to oranges in its simplest form?

Click to see Solution

1. Write the initial ratio: 12 : 18
2. Find the biggest number that divides into both 12 and 18, which is 6.
3. Divide both sides by 6: (12 ÷ 6) : (18 ÷ 6) = 2 : 3

Answer: 2 : 3

Question 2 (Easy • Colored Balls): Children are playing in a playground. The ratio of green plastic balls to yellow plastic balls is 4 : 12. Simplify this ratio.

Click to see Solution

1. Write the ratio: 4 : 12
2. Divide both numbers by their greatest common factor, which is 4.
3. (4 ÷ 4) : (12 ÷ 4) = 1 : 3

Answer: 1 : 3

Question 3 (Medium • Ice Cream Cups): At a school sports festival, vendors sold vanilla and chocolate ice cream cups in the ratio 3 : 5. If they sold 30 cups of vanilla ice cream, how many cups of chocolate ice cream did they sell?

Click to see Solution

1. The ratio of vanilla to chocolate is 3 parts to 5 parts.
2. Since 3 parts equal 30 actual cups, find the value of 1 part: 30 ÷ 3 = 10.
3. Multiply the chocolate parts by the same number: 5 × 10 = 50.

Answer: 50 cups of chocolate ice cream

Question 4 (Medium • Plates of Rice): A caterer cooked a total of 120 plates of rice for a wedding. The plates of Jollof rice and Fried rice are shared in the ratio 5 : 3. How many plates of Jollof rice were cooked?

Click to see Solution

1. Find the total number of parts combined: 5 + 3 = 8 parts.
2. Find the value of 1 part out of the total rice: 120 plates ÷ 8 = 15 plates.
3. Jollof rice takes 5 parts of the share: 5 × 15 plates = 75 plates.

Answer: 75 plates of Jollof rice

Question 5 (Hard • Mixed Party Items): An entertainment center sets up a kids' game arena with blue balls, red balls, and yellow balls in the ratio 2 : 3 : 5. If there are 45 red balls, what is the total number of plastic balls in the arena?

Click to see Solution

1. Red balls correspond to the middle part of the ratio, which is 3 parts. If 3 parts = 45 balls, then 1 part = 45 ÷ 3 = 15 balls.
2. Find the total parts in the game arena together: 2 + 3 + 5 = 10 parts.
3. Multiply total parts by the value of 1 part: 10 parts × 15 balls = 150 balls.

Answer: 150 total plastic balls

Calculating The Rate Of Doing A Piece Of Work By Different Number Of Men

 

Men and Days Problems: Worker Time Rates

In math, when more people join a job, the work gets done faster (it takes fewer days). If fewer people do the job, it takes longer (more days). This is called inverse relationship.

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The "Man-Days" Formula

The total amount of work needed to finish a job can be measured in "Man-Days" (Men × Days). Because the total work stays the same, we use this simple formula:

M₁ × D₁ = M₂ × D₂

• M₁ = Number of men at first
• D₁ = Number of days the first group takes
• M₂ = Number of men in the second group
• D₂ = Number of days the second group will take

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Practice Questions and Solutions (Ascending Difficulty)

Question 1: Very Simple (Doubling the Workers)

If 5 men can complete a piece of work in 20 days, how long will it take 10 men to do the same work?

Solution:

• First Group: M₁ = 5 men, D₁ = 20 days
• Second Group: M₂ = 10 men, D₂ = ?
• Apply Formula: 5 × 20 = 10 × D₂
• Calculate: 100 = 10 × D₂
  D₂ = 100 ÷ 10 = 10 days

Answer: It will take 10 men 10 days.

Question 2: Simple (Reducing the Workers)

If 6 builders can build a wall in 4 days, how many days will it take just 2 builders to build the same wall?

Solution:

• First Group: M₁ = 6, D₁ = 4
• Second Group: M₂ = 2, D₂ = ?
• Apply Formula: 6 × 4 = 2 × D₂
• Calculate: 24 = 2 × D₂
  D₂ = 24 ÷ 2 = 12 days

Answer: It will take 2 builders 12 days.

Question 3: Medium (Finding Number of Men)

A group of 8 farmers can clear a farmland in 6 days. How many farmers are needed to clear the same farmland in exactly 4 days?

Solution:

• First Group: M₁ = 8, D₁ = 6
• Second Group: M₂ = ?, D₂ = 4
• Apply Formula: 8 × 6 = M₂ × 4
• Calculate: 48 = M₂ × 4
  M₂ = 48 ÷ 4 = 12 farmers

Answer: 12 farmers are needed.

Question 4: Hard (More Men Joining Later)

12 men are hired to dig a trench, and they can finish it in 5 days. If 3 more men join the team before they start, how many days will it take them all together?

Solution:

1. Find the new number of men: They started with 12 men, and 3 more joined. So, M₂ = 12 + 3 = 15 men.
2. Identify values: M₁ = 12, D₁ = 5, M₂ = 15, D₂ = ?
3. Apply Formula: 12 × 5 = 15 × D₂
4. Calculate: 60 = 15 × D₂
   D₂ = 60 ÷ 15 = 4 days

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Calculating The Rate of Doing Work

 

Understanding "Rate of Doing Work" in Primary School Math

In primary school math, the rate of doing work tells us how much of a task (like painting a wall or filling a tank) someone or something can complete in a single unit of time, such as 1 hour, 1 day, or 1 minute.

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The Formula

To solve these problems, we use one simple formula:

Rate of Work = Total Work Done ÷ Time Taken

Important Rule: In math, a single complete job is always represented as 1 whole. For example, if a person takes 5 days to complete a job, their daily rate of work is 1/5 of that job.

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Practice Questions and Solutions (Ascending Difficulty)

Question 1: Very Simple (Direct Rate)

John can paint a fence in 4 hours. What fraction of the fence can he paint in just 1 hour?

Solution:

• Total Work: 1 whole fence
• Total Time: 4 hours
• Calculation: Rate = 1/4

Answer: John can paint 1/4 of the fence in 1 hour.

Question 2: Simple (Finding Total Time)

A water pump can fill 1/5 of a swimming pool in 1 hour. How many hours will it take the pump to fill the entire pool?

Solution:

• Rate of Work: 1/5 of the pool per hour
• Total Work: 1 whole pool
• Calculation: Total Time = 1 ÷ (1/5) = 1 × 5 = 5 hours

Answer: It will take the pump 5 hours to fill the entire pool.

Question 3: Medium (Working Together)

Mary can clean a classroom in 3 hours, and Jane can clean the same classroom in 6 hours. If they work together, what fraction of the classroom will they clean in 1 hour?

Solution:

• Mary's 1-hour rate: 1/3 of the room
• Jane

How To Calculate Area Of Triangles

 

Mathematics is full of fascinating shapes, but one of the most fundamental shapes you will ever encounter is the triangle. Whether you are looking at the roof of a house, a slice of pizza, or a yield traffic sign, triangles are everywhere!

In this post, we will define what a triangle is, explore its four primary types with visual diagrams, and solve 5 simple practice questions perfect for primary school students.

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What is a Triangle?

A triangle is a closed, three-sided geometric shape. It is formed by connecting three straight line segments. Every triangle has exactly:

• 3 sides (the straight edges)
• 3 vertices (the corner points where the sides meet)
• 3 internal angles (the space inside the corners)

Fun Fact: No matter how big or small a triangle is, the sum of its three internal angles always adds up to exactly 180°.

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The Four Main Types of Triangles

Triangles are generally classified by the lengths of their sides or the sizes of their angles. Here is how the four main types look and differ from one another:

Equilateral
All sides equal

Isosceles
2 sides equal

Scalene
No sides equal

Right-Angled
One 90° angle

1. Equilateral Triangle

An equilateral triangle is a triangle where all three sides are equal in length. Because the sides are equal, all three internal angles are also equal, each measuring exactly 60°.

2. Isosceles Triangle

An isosceles triangle has two sides of equal length and one side that is different. The angles opposite to the equal sides are also equal to each other.

3. Scalene Triangle

A scalene triangle is a triangle where all three sides have different lengths. Consequently, all three internal angles have different measurements as well.

4. Right-Angled Triangle

A right-angled triangle (or right triangle) is a triangle that has one internal angle that measures exactly 90° (called a right angle). The side opposite the right angle is always the longest side, known as the hypotenuse.

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How to Calculate the Area of a Triangle

To find out how much space is inside a triangle, you just need to know two measurements: the Base (the bottom side) and the Height (the straight vertical distance from the top point down to the base).

← ——— Base (B) ———→

The formula is:

Area = ½ × Base × Height

Or simply: Multiply the base by the height, and then divide the answer by 2.

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5 Simple Questions and Solutions for Primary School

Question 1

Find the area of a triangle with a base of 6 cm and a height of 4 cm.

Solution:

• Base = 6 cm, Height = 4 cm
• Area = ½ × Base × Height
• Area = ½ × 6 × 4
• Area = ½ × 24
• Answer: Area = 12 cm²

Question 2

A triangular biscuit has a base of 8 cm and a height of 5 cm. What is its area?

Solution:

• Base = 8 cm, Height = 5 cm
• Area = ½ × 8 × 5
• Area = ½ × 40
• Answer: Area = 20 cm²

Question 3

A right-angled triangle has a base of 10 cm and a height of 3 cm. Calculate its area.

Solution:

• Base = 10 cm, Height = 3 cm
• Area = ½ × 10 × 3
• Area = 5 × 3
• Answer: Area = 15 cm²

Question 4

The base of a triangle is 12 cm and its height is 6 cm. Find the area.

Solution:

• Base = 12 cm, Height = 6 cm
• Area = ½ × 12 × 6
• Area = 6 × 6
• Answer: Area = 36 cm²

Question 5

A small triangular flag has a height of 7 cm and a base of 4 cm. What is the area of the flag?

Solution:

• Base = 4 cm, Height = 7 cm
• Area = ½ × 4 × 7
• Area = 2 × 7
• Answer: Area = 14 cm²

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Parents and Teachers: Feel free to change the numbers in these questions to give your children more practice! Happy learning!

Conversion From Base Two To Base Ten

 

Converting Base Two to Base Ten

To convert a number from Base Two (Binary) back to Base Ten (Decimal), we multiply each digit by its positional power of 2 (starting from 2⁰ on the far right) and add the results together.

Question 1: Convert 110₂ to Base Ten

Solution:
• (1 × 2²) + (1 × 2¹) + (0 × 2⁰)
• (1 × 4) + (1 × 2) + (0 × 1)
• 4 + 2 + 0 = 6

Answer: 110₂ = 6₁₀

Question 2: Convert 1001₂ to Base Ten

Solution:
• (1 × 2³) + (0 × 2²) + (0 × 2¹) + (1 × 2⁰)
• (1 × 8) + (0 × 4) + (0 × 2) + (1 × 1)
• 8 + 0 + 0 + 1 = 9

Answer: 1001₂ = 9₁₀

Question 3: Convert 1100₂ to Base Ten

Solution:
• (1 × 2³) + (1 × 2²) + (0 × 2¹) + (0 × 2⁰)
• (1 × 8) + (1 × 4) + (0 × 2) + (0 × 1)
• 8 + 4 + 0 + 0 = 12

Answer: 1100₂ = 12₁₀

Question 4: Convert 1111₂ to Base Ten

Solution:
• (1 × 2³) + (1 × 2²) + (1 × 2¹) + (1 × 2⁰)
• (1 × 8) + (1 × 4) + (1 × 2) + (1 × 1)
• 8 + 4 + 2 + 1 = 15

Answer: 1111₂ = 15₁₀

Question 5: Convert 10100₂ to Base Ten

Solution:
• (1 × 2⁴) + (0 × 2³) + (1 × 2²) + (0 × 2¹) + (0 × 2⁰)
• (1 × 16) + (0 × 8) + (1 × 4) + (0 × 2) + (0 × 1)
• 16 + 0 + 4 + 0 + 0 = 20

Answer: 10100₂ = 20₁₀

Introduction To Base Two

 

Introduction to Number Bases (Base Two)

In our everyday lives, we use Base Ten (Decimal), which relies on ten digits (0 through 9). However, computers and digital systems don't understand ten digits—they operate on electricity, which is either on or off.

This is where Base Two (Binary) comes in. The binary system uses only two digits: 0 and 1.

• 0 represents "off" (low voltage).
• 1 represents "on" (high voltage).

How Base Two Works

Just like base ten uses powers of 10 (1, 10, 100, 1000), base two uses powers of 2. The place values double each time you move to the left:

24 (Sixteens)	23 (Eights)	22 (Fours)	21 (Twos)	20 (Ones)
16	8	4	2	1

To show a number is in a specific base, we write a small subscript. For example, 13₁₀ (thirteen in base ten) is written as 1101₂ in base two.

Converting Base Ten to Base Two

To convert a number from base ten to base two, we use the repeated division method. Divide the number by 2, write down the remainder, and repeat until you get 0. Then, read the remainders from the bottom to the top.

Question 1: Convert 6₁₀ to Base Two

Solution:
• 6 ÷ 2 = 3 remainder 0
• 3 ÷ 2 = 1 remainder 1
• 1 ÷ 2 = 0 remainder 1

Answer: 6₁₀ = 110₂

Question 2: Convert 9₁₀ to Base Two

Solution:
• 9 ÷ 2 = 4 remainder 1
• 4 ÷ 2 = 2 remainder 0
• 2 ÷ 2 = 1 remainder 0
• 1 ÷ 2 = 0 remainder 1

Answer: 9₁₀ = 1001₂

Question 3: Convert 12₁₀ to Base Two

Solution:
• 12 ÷ 2 = 6 remainder 0
• 6 ÷ 2 = 3 remainder 0
• 3 ÷ 2 = 1 remainder 1
• 1 ÷ 2 = 0 remainder 1

Answer: 12₁₀ = 1100₂

Question 4: Convert 15₁₀ to Base Two

Solution:
• 15 ÷ 2 = 7 remainder 1
• 7 ÷ 2 = 3 remainder 1
• 3 ÷ 2 = 1 remainder 1
• 1 ÷ 2 = 0 remainder 1

Answer: 15₁₀ = 1111₂

Question 5: Convert 20₁₀ to Base Two

Solution:
• 20 ÷ 2 = 10 remainder 0
• 10 ÷ 2 = 5 remainder 0
• 5 ÷ 2 = 2 remainder 1
• 2 ÷ 2 = 1 remainder 0
• 1 ÷ 2 = 0 remainder 1

Answer: 20₁₀ = 10100₂