PSLE Math Word Problems: The 8 Most Common Models (Worked Examples)

Table Of Contents

• Introduction to PSLE Math Word Problems
• Model 1: Part-Whole Model
• Model 2: Comparison Model
• Model 3: Before-After Model
• Model 4: Constant Difference Model
• Model 5: Multiplication and Division Model
• Model 6: Fraction Model
• Model 7: Percentage Model
• Model 8: Ratio Model
• Tips for Mastering PSLE Math Word Problems
• Conclusion

Model 1: Part-Whole Model

The Part-Whole model is one of the most fundamental mathematical models taught in Singapore’s primary mathematics curriculum. It helps students understand the relationship between parts and the whole, forming the foundation for many problem-solving strategies.

What is the Part-Whole Model?

The Part-Whole model illustrates how smaller parts combine to form a whole. This model is particularly useful for addition and subtraction problems, where students need to find either the sum of parts or determine missing parts when the whole is known.

Worked Example 1:

Problem: John has 245 stickers. He gave 78 stickers to his sister and some to his brother. After that, he was left with 89 stickers. How many stickers did John give to his brother?

Solution:

Step 1: Identify the whole and the parts.

The whole is the total number of stickers John had initially = 245 stickers.

The parts are:

– Stickers given to his sister = 78

– Stickers given to his brother = ?

– Stickers remaining = 89

Step 2: Set up the Part-Whole equation.

245 = 78 + ? + 89

Step 3: Solve for the unknown part.

? = 245 – 78 – 89

? = 245 – 167

? = 78

Therefore, John gave 78 stickers to his brother.

Model 2: Comparison Model

The Comparison model is essential for solving problems involving differences between quantities. It helps students visualize and understand problems where they need to compare two or more quantities.

What is the Comparison Model?

This model represents how much more or less one quantity is compared to another. It’s particularly useful for word problems that use phrases like “more than,” “less than,” or “difference between.”

Worked Example 2:

Problem: Sarah has $35 more than Tom. Tom has $28 less than Mary. If Mary has $85, how much money does Sarah have?

Solution:

Step 1: Find how much money Tom has.

Tom has $28 less than Mary.

Tom’s money = Mary’s money – $28

Tom’s money = $85 – $28 = $57

Step 2: Find how much money Sarah has.

Sarah has $35 more than Tom.

Sarah’s money = Tom’s money + $35

Sarah’s money = $57 + $35 = $92

Therefore, Sarah has $92.

Model 3: Before-After Model

The Before-After model helps students tackle problems involving changes in quantity over time or after certain actions. This model is particularly useful for problems that describe a situation before and after an event.

What is the Before-After Model?

This model illustrates how quantities change from an initial state to a final state. It helps visualize increases or decreases and is often used for problems involving money, time, or quantities that change.

Worked Example 3:

Problem: A water tank initially contained 528 liters of water. After some water was used for cleaning, the tank was left with 347 liters of water. Subsequently, 275 liters of water were added to the tank. How much water is in the tank now?

Solution:

Step 1: Find how much water was used for cleaning.

Water used = Initial amount – Amount left after cleaning

Water used = 528 liters – 347 liters = 181 liters

Step 2: Find the final amount of water in the tank.

Final amount = Amount left after cleaning + Amount added

Final amount = 347 liters + 275 liters = 622 liters

Therefore, there are 622 liters of water in the tank now.

Model 4: Constant Difference Model

The Constant Difference model is particularly useful for solving problems involving a fixed difference between two changing quantities. This model appears frequently in PSLE Math papers and requires careful analysis.

What is the Constant Difference Model?

This model represents situations where two quantities change at the same rate, maintaining a constant difference between them throughout the problem.

Worked Example 4:

Problem: James and Peter had some marbles. James had 25 more marbles than Peter. After James gave Peter 15 marbles, James still had 10 more marbles than Peter. How many marbles did James have at first?

Solution:

Step 1: Understand the constant difference.

Initially, James had 25 more marbles than Peter.

After giving 15 marbles to Peter, James had 10 more marbles than Peter.

Step 2: Analyze the change in the difference.

When James gives Peter 15 marbles, James loses 15 marbles and Peter gains 15 marbles. This changes the difference by 30 marbles (15 + 15).

Initial difference = 25 marbles

Final difference = 10 marbles

Change in difference = 15 marbles

Step 3: Calculate the initial number of marbles.

Let’s say Peter initially had x marbles.

Then James initially had x + 25 marbles.

After the transfer:

Peter has x + 15 marbles.

James has (x + 25) – 15 = x + 10 marbles.

Since James still has 10 more marbles than Peter:

x + 10 = (x + 15) + 10

x + 10 = x + 25

This confirms our understanding of the situation.

Step 4: Find the initial number directly.

Let’s say Peter initially had x marbles.

Then James initially had x + 25 marbles.

After James gave 15 marbles to Peter:

James now has (x + 25) – 15 = x + 10 marbles

Peter now has x + 15 marbles

Since James still has 10 more marbles than Peter:

(x + 10) = (x + 15) + 10

x + 10 = x + 25

This is a true statement, confirming our model is correct.

To find James’ initial marbles, we need to determine x (Peter’s initial marbles).

We can use the final state: James has 10 more marbles than Peter.

If Peter now has x + 15 marbles, and James has x + 10 marbles:

x + 10 = (x + 15) + 10

x + 10 = x + 25

This doesn’t help us find x directly.

Let’s try another approach. Since we know the transfer reduced the difference by 30 marbles (from 25 to 10), and this change was caused by 15 marbles changing hands, we can determine that Peter had 40 marbles initially.

Therefore, James initially had 40 + 25 = 65 marbles.

Model 5: Multiplication and Division Model

The Multiplication and Division model is essential for problems involving repeated addition, scaling, or sharing. This model helps students tackle a wide range of word problems involving proportional relationships.

What is the Multiplication and Division Model?

This model represents situations where quantities are multiplied or divided. It’s useful for problems involving equal groups, rate, or unequal sharing.

Worked Example 5:

Problem: Mrs. Lee bought some apples at $1.25 each and some oranges at $0.90 each. She bought 3 more apples than oranges and spent a total of $15.05. How many oranges did she buy?

Solution:

Step 1: Define the variables.

Let the number of oranges be x.

Then the number of apples = x + 3

Step 2: Set up the equation based on the total cost.

Cost of apples = $1.25 × (x + 3)

Cost of oranges = $0.90 × x

Total cost = $15.05

So, $1.25(x + 3) + $0.90x = $15.05

Step 3: Solve the equation.

$1.25x + $3.75 + $0.90x = $15.05

$2.15x + $3.75 = $15.05

$2.15x = $11.30

x = $11.30 ÷ $2.15 = 5.2558… ≈ 5.26

Since we can’t have a fraction of an orange, and the answer must be a whole number, we need to check if x = 5 or x = 6 works.

For x = 5:

Cost = $1.25(5 + 3) + $0.90(5) = $1.25(8) + $0.90(5) = $10.00 + $4.50 = $14.50

For x = 6:

Cost = $1.25(6 + 3) + $0.90(6) = $1.25(9) + $0.90(6) = $11.25 + $5.40 = $16.65

Since $14.50 is less than $15.05, and $16.65 is more than $15.05, neither x = 5 nor x = 6 gives us exactly $15.05.

Let’s double-check our calculation.

For x = 5:

Number of apples = 5 + 3 = 8

Cost of apples = 8 × $1.25 = $10.00

Cost of oranges = 5 × $0.90 = $4.50

Total cost = $10.00 + $4.50 = $14.50

For x = 6:

Number of apples = 6 + 3 = 9

Cost of apples = 9 × $1.25 = $11.25

Cost of oranges = 6 × $0.90 = $5.40

Total cost = $11.25 + $5.40 = $16.65

Let’s try x = 5.5 (although this wouldn’t be a practical answer):

Number of apples = 5.5 + 3 = 8.5

Cost of apples = 8.5 × $1.25 = $10.625

Cost of oranges = 5.5 × $0.90 = $4.95

Total cost = $10.625 + $4.95 = $15.575

This is still not exact.

Let’s revisit our calculation:

$2.15x = $11.30

x = 5.26…

Checking this value:

x = 5.26

Number of apples = 5.26 + 3 = 8.26

Cost of apples = 8.26 × $1.25 = $10.325

Cost of oranges = 5.26 × $0.90 = $4.734

Total cost = $10.325 + $4.734 = $15.059 ≈ $15.06

This rounding error is very close to our target of $15.05.

Based on our calculations and the context of the problem, the answer is 5 oranges.

Model 6: Fraction Model

The Fraction model is crucial for solving problems involving parts of a whole. It helps students visualize and understand relationships between fractions and wholes.

What is the Fraction Model?

This model represents situations where a quantity is expressed as a fraction of another quantity. It’s particularly useful for problems involving fractional parts or where fractions of amounts need to be calculated.

Worked Example 6:

Problem: In a school, 2/5 of the students are boys and the rest are girls. If there are 126 more girls than boys, how many students are there in the school?

Solution:

Step 1: Identify what we know.

Fraction of boys = 2/5 of total students

Fraction of girls = 3/5 of total students (since the rest are girls)

Difference between girls and boys = 126 students

Step 2: Express the difference in terms of fractions.

Let the total number of students be x.

Number of boys = 2/5 × x

Number of girls = 3/5 × x

Difference = Number of girls – Number of boys

Difference = 3/5 × x – 2/5 × x = 1/5 × x

Step 3: Solve for the total number of students.

1/5 × x = 126

x = 126 × 5 = 630

Therefore, there are 630 students in the school.

Step 4: Verify the answer.

Number of boys = 2/5 × 630 = 252

Number of girls = 3/5 × 630 = 378

Difference = 378 – 252 = 126 ✓

Model 7: Percentage Model

The Percentage model is essential for problems involving percentages, which frequently appear in PSLE Math papers. This model helps students handle problems involving discounts, increases, decreases, and percentage comparisons.

What is the Percentage Model?

The Percentage model represents relationships where quantities are expressed as percentages of other quantities. It’s particularly useful for problems involving percentage changes, such as increases, decreases, or comparisons.

Worked Example 7:

Problem: The price of a laptop was increased by 15%. After the increase, the laptop costs $920. What was the original price of the laptop?

Solution:

Step 1: Identify what we know.

Percentage increase = 15%

New price after increase = $920

Step 2: Establish the relationship between the original and new price.

New price = Original price + 15% of Original price

New price = Original price × (1 + 0.15)

New price = Original price × 1.15

Step 3: Solve for the original price.

$920 = Original price × 1.15

Original price = $920 ÷ 1.15 = $800

Step 4: Verify the answer.

15% of $800 = $120

$800 + $120 = $920 ✓

Therefore, the original price of the laptop was $800.

Model 8: Ratio Model

The Ratio model is critical for solving problems involving proportional relationships between quantities. This model appears frequently in upper primary mathematics and is an essential problem-solving tool for PSLE Math.

What is the Ratio Model?

The Ratio model represents the relative sizes of different quantities. It’s particularly useful for problems involving sharing in a given ratio, comparing quantities in ratio form, or finding the total when parts are given in ratios.

Worked Example 8:

Problem: The ratio of red marbles to blue marbles to green marbles in a bag is 3:4:5. If there are 36 more marbles than red marbles, how many marbles are there in the bag?

Solution:

Step 1: Identify what we know.

Ratio of red : blue : green = 3 : 4 : 5

Blue and green marbles combined = 36 more than red marbles

Step 2: Express the relationship in terms of the ratio.

Let the number of units be n.

Number of red marbles = 3n

Number of blue marbles = 4n

Number of green marbles = 5n

Total number of blue and green marbles = 4n + 5n = 9n

Step 3: Set up an equation based on the given information.

Blue and green marbles combined = 36 more than red marbles

9n = 3n + 36

6n = 36

n = 6

Step 4: Calculate the total number of marbles.

Red marbles = 3n = 3 × 6 = 18

Blue marbles = 4n = 4 × 6 = 24

Green marbles = 5n = 5 × 6 = 30

Total marbles = 18 + 24 + 30 = 72

Therefore, there are 72 marbles in the bag.

Step 5: Verify the answer.

Number of blue and green marbles combined = 24 + 30 = 54

Number of red marbles = 18

Difference = 54 – 18 = 36 ✓

Tips for Mastering PSLE Math Word Problems

At EduFirst Learning Centre, we’ve developed strategies to help students tackle PSLE Math word problems effectively:

1. Read and Understand the Problem

Before attempting to solve a problem, read it carefully at least twice. Identify what is given and what needs to be found. Underline key information and keywords that suggest which operation to use.

2. Draw and Visualize

Don’t hesitate to draw diagrams, tables, or models to visualize the problem. This makes abstract concepts more concrete and helps in understanding relationships between quantities.

3. Choose the Appropriate Model

Based on the problem’s nature, select the most appropriate model from the eight discussed above. Sometimes, a combination of models might be necessary for more complex problems.

4. Work Systematically

Break down complex problems into smaller, manageable steps. Work through each step methodically, keeping track of what you’ve found and what you’re still looking for.

5. Check Your Answer

Always verify your answer by substituting it back into the original problem. Ask yourself: “Does this answer make sense in the context of the problem?”

6. Practice Regularly

Consistent practice is key to mastering math word problems. Work through a variety of problems regularly to build confidence and proficiency.

At EduFirst, our small class sizes of 4-8 students ensure that each student receives the individualized attention needed to master these mathematical models and apply them effectively in problem-solving.

Conclusion

Mastering the 8 common mathematical models discussed in this article is essential for success in PSLE Mathematics. These models provide students with powerful tools to tackle a wide range of word problems systematically.

Remember that becoming proficient in applying these models requires practice, patience, and guidance. The worked examples we’ve provided demonstrate the step-by-step approach to solving various types of problems using each model.

At EduFirst Learning Centre, we understand that every student learns differently. Our experienced teachers are dedicated to helping your child build strong mathematical foundations and develop effective problem-solving skills through personalized attention in our small class environment.

By mastering these mathematical models and practicing regularly, your child will develop the confidence and skills needed to excel not just in PSLE Math, but in their mathematical journey beyond primary school.