Whether you’re taking the UPCAT, Civil Service Exam, NMAT, or other written examinations, word problems will always be part of the Mathematics subtest. They’re among the most frequently asked questions and also one of the most challenging.

This section will provide some tips on approaching these word problems using algebra. In particular, we will solve number problems, age problems, work problems, rate problems, and so on.

Hopefully, you can answer word problems with ease after reading this chapter.

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General Tips in Solving Word Problems Using Algebra

Here are some quick tips to help you improve your problem-solving skills in mathematics:

1. Read the problem carefully and determine what is being asked. You must know what you must solve because it will lead you to the process you must perform to answer the problem.
2. Identify the given and define them. Identify all quantities that are provided in the problem. Ensure you know these quantities and how they relate to the problem.
3. Set up a mathematical expression that represents the problem. Your mathematical expression can be an equation or inequality. You must make the correct expression to obtain the precise answer.
4. Solve the mathematical expression you have made. Ensure that you know the techniques on how to solve equations and inequalities. If you are not confident enough with your arithmetic and algebra of equations and inequalities, practicing them before attempting to answer these word problems is highly suggested. 

Keep the above-mentioned things in mind as we discuss how to solve different word problems. 

 

Number Problems

Number problems are word problems that ask you to determine a specific number or quantity using the descriptions given in the problem. 

To solve number problems, you should be familiar with some keywords used in algebra. These keywords, such as sum, increased by, decreased by, of, and ratio of allow you to determine what mathematical operation/s is involved in the problem.

For instance, a number problem containing the statement “the sum of a number and 5” gives a clue that the problem involves adding numbers.

Let us try to solve some number problems:

Sample Problem 1

Twice a number increased by five is 125. What is the number?

Solution:

The problem above is asking for an unknown number.

Let x be that unknown number.

Recall that the keyword “twice” means “double” or a certain number is multiplied by 2. Thus, twice the unknown number is 2x.

Twice that unknown number increased by five can be represented by 2x + 5.

Therefore, our equation would be 2x + 5 = 125. If we solve for x in this equation, we can determine the unknown number in the problem.

Solving the equation:

2x + 5 = 125

2x  = -5 + 125 Transposition method

2x = 120

(2x)/2 = 120/2 Dividing both sides of the equation by 2

x = 60

Since x represents the unknown number in the problem, and we have computed that x = 60, the unknown number is 60.

Sample Problem 2

When a number is increased by its reciprocal, the result is 2. Determine the number.

Solution:

The problem is asking for an unknown number.

Let x be that unknown number.

As stated in the problem, if the number is increased by its reciprocal, we will obtain 2. The reciprocal of x is 1/x. Thus, our equation would be:

x + 1/x = 2

To solve this equation, we must first eliminate the denominator. This can be achieved by multiplying both sides of the equation by the Least Common Denominator (LCD), which is x:

x(x + 1/x) = x(2)

x² + 1  = 2x

Notice that after we eliminate the denominator, the resulting equation is quadratic. We can write it in standard form as follows:

x² + 1 = 2x

x² – 2x + 1 = 0 Transposition method

Now, let us solve for x in x² – 2x + 1 = 0 to determine the unknown number:

x² – 2x + 1 = 0 

(x – 1)(x – 1) = 0 By factoring

x – 1 = 0 x – 1 = 0 Set each factor to 0

x = 1 x = 1

Based on our solution, the unknown number in the problem is 1. 

You can also verify that the sum of 1 and its reciprocal is 2. The reciprocal of 1 is just 1. Hence, 1 + 1 = 2. 

Sample Problem 3

The sum of four consecutive positive whole numbers is 78. Determine the largest number among the consecutive numbers.

Solution:

Again, this problem is asking for an unknown number. 

Consecutive numbers are numbers that follow each other continuously in order from smallest to largest. For example, 8, 9, 10, and 11 are consecutive numbers.

Now, the problem asks us to find four consecutive positive whole numbers whose sum is 78. 

Let x represent the smallest or the first number of these four consecutive numbers.

x – smallest/first number

Since the numbers are consecutive, the second number of these four consecutive numbers must be x + 1 since you have to add 1 to the first number (which is x) to obtain the next number.

x + 1 – second number

Now, we represent the third number as x + 2 since we have to add 2 to the first number (which is x) to obtain the third number.

x + 2 – third number

Lastly, we represent the fourth number or the largest number as x + 3 since we have to add 3 to the first number (which is x) to obtain the fourth or the largest number.

x + 3 – fourth number

So far, we have the following to represent the four consecutive numbers:

x – first number

x + 1 – second number

x + 2 – third number

x + 3 – fourth number

Recall that the sum of these four consecutive numbers is 78. Thus, we have this equation:

first number + second number + third number + fourth number = 78

x + (x + 1) + (x + 2) + (x + 3) = 78

We can simplify the equation above as follows:

4x + 6 = 78

Let us solve for the value of x in the equation above:

4x + 6 = 78

4x = -6 + 78 Transposition method

4x = 72

(4x)/4 = 72/4 Dividing both sides of the equation by 4

x = 18

Since x represents the smallest or the first number and we have obtained x = 18, the first number should be 18. It follows that the second number is 19, the third number is 20, and the fourth or the largest number is 21. 

Therefore, the answer to the problem is 21.

 

Age Problems

One of the most commonly asked questions in mathematics proficiency exams involves age problems. These problems require you to determine the specific age of a person using details given in the problem.

Age problems are usually solvable using your intuition alone and without the need to apply algebraic concepts. However, relying on your intuition is not a reliable way to solve age problems and can be time-consuming. Furthermore, solutions obtained through this process tend to be unorganized, making it difficult to answer the given problem.

Using algebra, you can use a more systematic approach to solving age problems in less time.

Sample Problem 1

Letty is three times as old as Bert. The sum of Bert’s and Letty’s ages is 48. How old is Letty?

Solution:

The problem is asking for the age of Letty, given that she is three times older than Bert.

Let x be Bert’s age.

Since Letty is three times as old as Bert, we can write Letty’s age as 3x.

The sum of Bert’s and Letty’s ages is 48. Thus, our equation would be:

Bert’s age + Letty’s age = 48

x + 3x = 48

4x = 48

Using the equation, we have to solve for x to determine Bert’s age:

4x = 48

(4x)/4 = 48/4 Dividing both sides of the equation by 4

x = 12

Since x represents Bert’s age, then Bert is 12 years old. Now, we know that Letty is three times older than Bert. Therefore, Letty’s age is 3(12) = 36.

Letty is 36 years old. 

Sample Problem 2

This year, Albert is 11 years older than Isaac. Three years ago, the sum of their ages was 67. Determine how old Isaac and Albert are this year.

Solution:

The problem is asking for Albert’s and Isaac’s ages in the current year.

Let x be Isaac’s age in the present year.

The problem states that Albert is 11 years older this year than Isaac. Hence, we can represent Albert’s age as x + 11 this year.

This means that three years ago, Isaac’s age can be represented by x – 3. On the other hand, Albert’s age can be represented by x + 11 – 3 = x + 8.

To summarize:

	Present	Past (3 years ago)
Albert	x	x – 3
Isaac	x + 11	x + 8
Sum		67

The sum of the ages of Albert and Isaac in the past is 67. Hence, we can set up our equation as follows:

Albert’s age (past) + Isaac’s age (past) = 67

(x – 3) + (x + 8) = 67

We can simplify the equation above as follows:

2x + 5 = 67

Solving for x in the equation above:

2x + 5 = 67

2x = -5 + 67 Transposition method

2x = 62

2x/2 = 62/2 Dividing both sides of the equation by 2

x = 31

Since x represents Isaac’s age in the present, then Isaac is 31 years old in the present. Meanwhile, since Albert is 11 years older than Isaac in the present, Albert is 31 + 11 = 42 years old.

Hence, Isaac is 31 years old, while Albert is 42 years old.

Sample Problem 3

Five years from now, the sum of Richard’s age and Mike’s age will be 21. In the current year, Richard’s age is five years higher than twice Mike’s age. Determine Mike’s age in the present.

Solution:

The problem is asking for Mike’s age in the present.

It implies that Mike is younger than Richard. Hence, we let x be Mike’s age in the present.

Richard’s age is 5 years higher than twice Mike’s age. Twice of Mike’s age can be represented by 2x. Thus, Richard’s age in the present can be represented by 2x + 5.

Five years from now, Mike’s age will be x + 5 while Richard’s age will be 2x + 10.

To summarize:

	Present	Future (five years from now)
Mike’s age	x	x + 5
Richard’s age	2x + 5	2x + 10
Sum		21

The problem states that in the future (5 years from now), Mike’s and Richard’s ages will sum up to 21.

Thus, we have this equation:

Mike’s age (future) + Richard’s age (future) = 21

(x + 5) + (2x + 10) = 21

Combining like terms in the equation above:

3x + 15 = 21

Solving for x in the equation above:

3x + 15 = 21

3x = -15 + 21 Transposition method

3x = 6

(3x)/3 = 6/3 Dividing both sides of the equation by 3

x = 2

Looking back at our table of Richard’s and Mike’s ages, x represents Mike’s age in the present. Therefore, Mike is two years old at present.

Sample Problem 4

This year, Ravel is 12 years younger than Claude. Twelve years ago, Claude’s age was five times Ravel’s age. Determine how old Claude is in the present.

Solution:

The problem is asking for Claude’s age in the present.

Ravel seems younger than Claude, so we let x be Ravel’s age in the present.

Ravel is 12 years younger than Claude. This also means that Claude is 12 years older than Ravel. Hence, let x + 12 represent Claude’s age.

Ravel’s age 12 years ago can be represented by x – 12. On the other hand, Claude’s age can be represented by x + 12 – 12 = x.

To summarize:

	Present	Past (12 years ago)
Ravel’s age	x	x – 12
Claude’s age	x + 12	x

The problem states that 12 years ago, Claude’s age was five times Ravel’s age. 

Claude’s age = 5(Ravel’s age)

x = 5(x – 12) 

We can simplify the equation above as follows:

x = 5(x – 12) 

x = 5x – 60 Distributive Property

Let us solve the equation above:

x = 5x – 60

x – 5x = -60 Transposition method

-4x = -60

(-4x)/-4 = (-60)/(-4) Dividing both sides by -4

x = 15

Looking back at the table, x represents Ravel’s present age while x + 12 represents Claude’s present age. Using the value of x we have obtained from the solution (which is x = 15):

x + 12

(15) + 12 = 27

Thus, Claude is 27 years old today.

 

Work Problems

Solving a work problem involves figuring out how long a person or group will take to finish a particular task (or work) using the details provided in the problem.

We can use a guiding formula or equation when solving work problems. This equation tells us that the sum of the reciprocal of the time it takes for A and B to finish a task is equal to the reciprocal of the time it takes for A and B to work together to finish the same task.

Let us try to solve some work problems to see how this equation/formula works:

Sample Problem 1

Jacques can finish cleaning a garage in 2 hours. On the other hand, Kath can finish cleaning the same garage in 5 hours. Suppose Jacques and Kath work together to clean the same garage; how long would it take them to accomplish it?

Solution:

The problem is asking for the time it would take for Jacques and Kath to finish cleaning the garage if they work together.

Jacques can finish cleaning the garage in 2 hours.

Meanwhile, Kath can finish cleaning the garage in 5 hours.

We let t represent the time it would take for Jacques and Kath to finish cleaning the garage if they worked together.

Recall that in a work problem, we can obtain the time it would take for two persons to finish a certain task by solving this equation:

Therefore, to solve the problem, we can use the same equation/formula:

Using the values given in the problem:

Solving for the value of t:

½ + ⅕ = 1/t 

To make the equation above much easier to solve, we multiply both sides of the inequality by the LCD (the LCD is 10t):

10t(½ + ⅕) = 10t(1/t) 

5t + 2t = 10

Now, let us solve the resulting linear equation above to determine the value of t:

5t + 2t = 10

7t = 10

(7t)/7 = 10/7 Dividing both sides of the equation by 7

t = 10/7 

Based on our computation above, it would take 10/7 or 1 3/7 hours to completely clean the garage if Jacques and Kath decide to work together.

Sample Problem 2

There are two pipes in a large water tank: Pipe C and Pipe D. Pipe C puts water into the tank while Pipe D spills water. If only Pipe C is opened, it will fill the tank in 2 hours. Meanwhile,  if only Pipe D is opened, it will empty the tank in 3 hours. Suppose that for a particular purpose, both Pipe C and Pipe D are opened simultaneously; how many hours will it take for the tank to be full?

Solution:

The problem is asking how long it would take for Pipe C to fill the tank, given that Pipe D is also open.

We know Pipe C puts water into the tank while Pipe D spills water. This means that we should modify our formula for the work problem as follows:

We made the operation in the formula a subtraction sign because Pipe D is spilling water, so it would take Pipe C longer to fill the tank.

Let t be the time it would take for the tank to be full given that Pipe C and Pipe D are opened at the same time.

It takes 2 hours for Pipe C to fill the tank and 3 hours for Pipe D to empty it. Inputting these values into the equation:

½  – ⅓ = 1/t 

Now, let us solve for the value of t:

½ – ⅓ = 1/t

We multiply both sides of the equation by 6t to remove the denominators in the equation:

6t(½ – ⅓) = 6t(1/t)

3t – 2t = 6

t = 6

Based on our computation above, it would take 6 hours for the water tank to be full if both Pipe C and Pipe D were opened simultaneously.

 

Rate Problems

Rate problems refer to word problems that involve moving objects. Most of the rate problems will ask you about the distance a particular moving object covers (distance), how long it would take for a moving object to reach a certain point (time), or how fast the moving object is (rate or speed). 

Since we are dealing with a moving object’s distance, rate, and time, we must know the distance formula for moving objects.

Distance Formula

Distance = Rate x Time

The distance formula tells us that a moving object (e.g., car, bus, airplane, cyclist, etc.) will cover a distance equivalent to the product of the rate or speed of the object and the time or the duration of the object’s movement.

For example, if a car travels at a rate of 60 kph for 2 hours, the distance that the car will cover is:

Distance = Rate x Time

Distance = 60 x 2

Distance = 120 km

Remember the distance formula because we will use it a lot to solve rate problems.

Sample Problem 1

Cars A and B leave two different cities 400 km apart. Car A and B move toward each other at 60 and 50 kph, respectively. How many hours will it take for these two cars to meet or pass each other?

Solution:

Given their respective rates, the problem is asking how many hours it will take for Cars A and B to meet.

It is helpful to create an illustration for this problem:

Based on our image above, the total distance covered by the cars equals 400 km. 

Let t be the number of hours it will take for the cars to meet or pass each other.

We can create a table so that we can summarize the details of the given situation.

	Rate (in kph)	Time	Distance (Rate x Time)
Car A	60	t	60t
Car B	50	t	50t
Total			400

Based on our table above, the respective distances covered by both cars are 60t and 50t.

Again, as shown in the illustration, the total distance they will have covered (the sum of their respective distances at time t) by the time the two cars meet is 400 km.

Thus, our equation would be:

60t + 50t = 400

110t = 400

(110t)/110 = 400/110

t = 40/11 or 3.64 

Hence, the cars will meet after about 3.64 hours.

Sample Problem 2

Fred and Ludwig love cycling. One day they decided to test how long it would take to be 35 km apart if they started at a certain point and moved away from each other. Fred moves at a rate of 15 kph while Ludwig moves at 20 kph. Compute how long it would take for them to be 35 km apart.

Solution:

Given their respective rates, The problem asks for the number of hours it will take for Fred and Ludwig to be 35 km apart.

If we try to illustrate the problem above, it will look like this:

It is seen that the total distance covered by Fred and Ludwig is equal to 35 km if they are already 35 km apart from each other.

Given their respective rates, let t represent the time it takes for Ludwig and Fred to be 35 km apart.

To summarize:

	Rate	Time	Distance (Rate x Time)
Fred	15	t	15t
Ludwig	20	t	20t
Total			35

Based on our table above, the respective distances covered by Fred and Ludwig to be 35 km apart are 15t and 20t.

The sum of the distances covered by Fred and Ludwig when they are 35 km apart is also equal to 35 km. Thus, we have this equation:

15t + 20t = 35

35t = 35

t = 1

Given their respective rates, it will only take 1 hour for Fred and Ludwig to be 35 km apart.

 

Other Word Problems

To enrich your arsenal of problem-solving techniques, let us try to answer some word problems that don’t belong to any of the categories we have discussed.

Problem 1: A bag in a department store has been tagged with a 20% discount. If you buy the bag with the discount, you only have to pay ₱960. What is the original price of the bag?

Solution:

We are tasked to find the original price of the bag.

Let x be the bag’s original price (i.e., the bag’s price without the discount).

The discount amount (i.e., the amount to be deducted from the original price) can be represented by 0.20x.

The problem states that if the bag was bought with a discount, you only have to pay ₱960. Hence, ₱960 is the discounted price of the bag.

Note that

Original Price – Discount Amount = Discounted Price

Hence, our equation would be:

x – 0.20x = 960

0.80x = 960

Dividing both sides of the equation by 0.80:

(0.80x)/0.80 = 960/0.80

x = 1200

Therefore, the original price of the bag (price without the discount) is ₱1200.

Problem 2: You are planning to buy flower pots from an online shopping site as a gift to your mother. Each flower pot costs ₱350, and you have to pay a fixed amount of ₱100 as a delivery fee. What is the maximum number of flower pots you can buy, given that you only have a ₱1000 budget?

Solution:

The problem is asking for the largest number of flower pots your budget of ₱1000 can buy.

Let x be the number of flower pots.

Then, let 350x represent the total cost of buying x flower pots.

Since there’s a fixed service fee of ₱100, the total amount you must pay to the online shopping site would be 350x + 100.

Now, 350x + 100 must be less than or equal to 1000 since our budget is only ₱1000. In other words, the total amount we need to pay to the online shopping site should not exceed 1000.

Hence, we have this inequality:

350x + 100 ≤ 1000

Let us solve the inequality above:

350x ≤ -100 + 1000

350x ≤ 900

(350x)/350 ≤ 900/350

x ≤ 900/350

x ≤ 2.57

We write our answers in decimals to make our interpretation easier to understand.

The solution set of the inequality is x ≤ 2.57 or the set of all numbers less than or equal to 2.57. This means that the number of flower pots you can buy with your ₱1000 budget can be any number less than 2.57. 

However, we are looking for the maximum number of flower pots we can buy. Note that the number of flower pots should be a whole number (a fractional or decimal number of flower pots does not make sense). Since our solution set is x ≤ 2.57 or the set of all numbers less than or equal to 2.57 and the largest whole number in this set is 2, the maximum number of flower pots we can buy is 2.

The answer to this problem is two flower pots.

Problem 3: Vince bought a shirt from a supermarket in his town. He also bought a headset that cost ₱700, lower than twice the price of the shirt. The total amount Vince paid for the shirt and the headset is ₱1100. How much is the shirt that Vince bought?

Solution:

We are tasked to find the price of the shirt that Vince bought.

Let x be the price of the shirt.

Since the headset price is ₱700 lower than twice the price of the shirt, we can represent the headset price as 2x – 700.

The total amount that Vince paid for the shirt and headset was ₱1100:

Price of the shirt + Price of the headset = 1100

x + (2x – 700) = 1100

Let us solve the equation above:

3x – 700 = 1100

3x = 700 + 1100 Transposition method

3x = 1800

(3x)/3 = 1800/3 Dividing both sides of the equation by 3

x = 600

Hence, the price of the shirt is ₱600.

 

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