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Linear Equation Word Problems

Problem: The sum of three consecutive odd numbers is $63$. Find the largest odd number.

Solution:

1. Let the first odd number be $x$.

Since consecutive odd numbers differ by $2$, the next ones are $x + 2$ and $x + 4$.

2. Create equation (sum is $63$):

3. The three numbers are $19$, $21$, and $23$.

Answer: The largest odd number is 23.

Problem: Currently, a father is $3$ times as old as his son. In $5$ years, the sum of their ages will be $66$. How old is the son now?

Solution:

1. Let the son's current age be $x$. Thus, the father's current age is $3x$.

In $5$ years, the son will be $x + 5$ and the father will be $3x + 5$.

2. Create equation from future sum:

Answer: The son is currently 14 years old.

Problem: A rectangle is $5$ meters longer than it is wide. If its perimeter is $50$ meters, find its width.

Solution:

1. Let the width be $x$ meters. The length is $x + 5$ meters.

Perimeter formula is $2 \times (\text{width} + \text{length})$.

2. Create equation:

Answer: The width of the rectangle is 10 meters.

Problem: A piggy bank contains 10-baht and 5-baht coins, totaling $20$ coins. If the total value is $160$ baht, how many 10-baht coins are there?

Solution:

1. Let the number of 10-baht coins be $x$.

The number of 5-baht coins is $20 - x$.

Value of 10-baht coins $= 10x$, Value of 5-baht coins $= 5(20 - x)$.

2. Create equation from total value:

Answer: There are 12 10-baht coins.

Problem: Two-thirds of a certain number is $18$. Find the number.

Solution:

1. Let the number be $x$.

2. Create equation:

Answer: The number is 27.

Problem: John is $5$ years older than Mary. $3$ years ago, John was twice as old as Mary. How old is Mary now?

Solution:

1. Let Mary's current age be $x$. John's current age is $x + 5$.

$3$ years ago, Mary was $x - 3$, and John was $(x + 5) - 3 = x + 2$.

2. Create equation from the past condition:

Answer: Mary is currently 8 years old.

Problem: You bought adult tickets for $100$ baht each and child tickets for $50$ baht each. A total of $10$ tickets cost $700$ baht. How many adult tickets did you buy?

Solution:

1. Let the number of adult tickets be $x$. The number of child tickets is $10 - x$.

2. Create equation from total cost:

Answer: You bought 4 adult tickets.

Problem: Two cars start from the same point and travel in opposite directions. The first car travels at 60 km/h and the second travels at 80 km/h. In how many hours will they be 420 km apart?

Solution:

1. Let $t$ be the time traveled (hours).

Distance = Speed $\times$ Time. Since they travel in opposite directions, Total Distance = Distance 1 + Distance 2.

2. Create equation (total distance is 420 km):

3. Time $t = 3$.

Answer: They will be 420 km apart in 3 hours.

Problem: Nat leaves home on a bicycle traveling at 15 km/h. Two hours later, his father drives a car along the same route at 45 km/h to catch up. How many hours will it take the father to catch Nat?

Solution:

1. Let $t$ be the father's driving time (hours).

Nat's total cycling time = $t + 2$ hours.

2. Create equation since catching up means their distances traveled are equal:

3. Time $t = 1$.

Answer: It will take the father 1 hour to catch up with Nat.

Problem: Ake can finish a job in 4 hours, and Boy can finish the same job in 6 hours. If they work together, how many hours will it take them to finish the job?

Solution:

1. Let $x$ be the time they work together (hours).

Ake's work rate = $\frac{1}{4}$ of the job per hour. Boy's work rate = $\frac{1}{6}$ of the job per hour.

2. Create equation since the sum of work done by both in $x$ hours equals 1 (1 complete job):

3. Time $x = 2.4$.

Answer: They will take 2.4 hours (or 2 hours and 24 minutes) together.

Problem: A merchant has 20 liters of a 10% sugar syrup. How many liters of a 40% sugar syrup must be added to obtain a 25% sugar syrup mixture?

Solution:

1. Let $x$ be the volume of 40% syrup to add (liters).

Amount of sugar in part 1 = $0.10 \times 20$. Amount of sugar in part 2 = $0.40 \times x$.

2. Create equation since total sugar must equal sugar in the new mixture = $0.25 \times (20 + x)$:

3. Syrup volume $x = 20$.

Answer: 20 liters of the 40% syrup must be added.

Problem: A store offers a 20% discount on an item, allowing a customer to buy it for 1,200 THB. What was the original price of the item?

Solution:

1. Let $x$ be the original price (THB).

A 20% discount means it is sold at 80% of the original price.

2. Create equation since 80% of $x$ equals 1,200 THB:

3. Original price $x = 1500$.

Answer: The original price of the item was 1,500 THB.

Problem: Two numbers are in the ratio 3 : 5. If 10 is added to both numbers, the new ratio becomes 5 : 7. Find the smaller number.

Solution:

1. Let the original numbers be $3x$ and $5x$.

After adding 10, they become $3x + 10$ and $5x + 10$.

2. Create equation since the new ratio is $\frac{5}{7}$:

3. We get $x = 5$, so the smaller number is $3x = 3(5) = 15$.

Answer: The smaller number is 15.

Problem: The sum of 3 consecutive even integers is 84. Find the largest integer.

Solution:

1. Let $x$ be the first even integer.

The next consecutive even integers are $x + 2$ and $x + 4$.

2. Create equation since their sum is 84:

3. We get $x = 26$. The largest is $x + 4 = 26 + 4 = 30$.

Answer: The largest integer is 30.

Problem: A positive two-digit number has a sum of its digits equal to 10. If the digits are reversed, the new number is 36 more than the original number. Find the original number.

Solution:

1. Let $x$ be the tens digit. Thus, the units digit is $10 - x$.

Original number value = $10x + (10 - x)$. Reversed number value = $10(10 - x) + x$.

2. Create equation since (Reversed - Original = 36):

3. The tens digit is 3, units digit is 7.

Answer: The original number is 37.

Problem: A mother is currently 3 times as old as her child. In 10 years, she will be twice as old as her child. How old is the child currently?

Solution:

1. Let the child's current age be $x$ years. Thus, the mother's current age is $3x$ years.

In 10 years, the child will be $x + 10$ years old, and the mother will be $3x + 10$ years old.

2. Create equation since (in 10 years, the mother's age will be twice the child's age):

3. We get $x = 10$.

Answer: The child is currently 10 years old.

Problem: A teacher bought pens for 15 THB each and pencils for 8 THB each, purchasing a total of 40 items for 460 THB. How many pens did the teacher buy?

Solution:

1. Let $x$ be the number of pens. Thus, the number of pencils is $40 - x$.

Cost of pens = $15x$ THB, and Cost of pencils = $8(40 - x)$ THB.

2. Create equation since the total amount spent is 460 THB:

3. We get $x = 20$.

Answer: The teacher bought 20 pens.

Problem: In a triangle, the second angle is twice the first angle, and the third angle is 20 degrees larger than the first angle. Find the measure of the first angle.

Solution:

1. Let the first angle measure $x$ degrees.

The second angle = $2x$ degrees, and the third angle = $x + 20$ degrees.

2. Create equation since the sum of interior angles of a triangle is 180 degrees:

3. We get $x = 40$.

Answer: The first angle measures 40 degrees.

Problem: A piggy bank contains 5-baht and 10-baht coins with a total value of 200 THB. If there are twice as many 5-baht coins as 10-baht coins, how many 10-baht coins are there?

Solution:

1. Let $x$ be the number of 10-baht coins.

Thus, the number of 5-baht coins is $2x$. Value of 10-baht coins = $10x$ THB, and value of 5-baht coins = $5(2x)$ THB.

2. Create equation since the total value is 200 THB:

3. We get $x = 10$.

Answer: There are 10 10-baht coins in the piggy bank.

Problem: A fraction has a denominator that is 5 more than its numerator. If 3 is added to both the numerator and the denominator, the new fraction equals $\frac{1}{2}$. Find the original fraction.

Solution:

1. Let the numerator be $x$. Thus, the denominator is $x + 5$ (original fraction is $\frac{x}{x + 5}$).

Adding 3 to both yields $\frac{x + 3}{(x + 5) + 3}$.

2. Create equation since the new fraction equals $\frac{1}{2}$:

3. The numerator is $x = 2$ and the denominator is $2 + 5 = 7$.

Answer: The original fraction is $\frac{2}{7}$.