## Math 8 Chapter 3 Lesson 7: Solve problems by making equations (continued)

## 1. Theoretical Summary

### 1.1. Representing a quantity by an expression containing the implicit

In practice, many variable quantities are interdependent. If one of these quantities is denoted by x, the other quantities can be expressed as an expression of the variable x.

### 1.2. Steps to solve the problem

**Step 1: **Equation

- Select unknowns and set appropriate conditions for unknowns
- Express unknown quantities in terms of unknowns and known quantities.
- Write an equation that shows the relationship between the quantities

**Step 2: **Solve the equation

**Step 3:** Reply. Check to see which of the solutions of the equation satisfy the condition of the unknowns and which do not and then draw conclusions.

## 2. Illustrated exercise

### 2.1. Exercise 1

At 6 o’clock, a motorbike leaves from A to arrive at B. After 1 hour, a car also leaves from A to B with an average speed 20km/h greater than the average speed of motorbikes. Both cars arrive at B at the same time at 9:30 on the same day. What is the length of the distance AB and the average speed of the motorcycle?

**Solution guide**

Let x (km) be the distance AB (x > 0).

Time to move from A to B of the motorbike:

9:30 – 6h = 3h30 = \( \frac{7}{2}\) (hour)

Speed of motorcycle: \(x \div \frac{7}{2}= \frac{2x}{7}\) (km/h)

Time to travel from A to B of the car: \( \frac{7}{2} – 1 = \frac{5}{2}\) (hours)

Speed of the car: \(x \div \frac{5}{2}= \frac{2x}{5}\)

Since the speed of the car is 20km/h more than the speed of the motorbike, we have the equation:

\( \frac{2x}{5} – \frac{2x}{7}= 20 \)

\(\Leftrightarrow 14x – 10x = 700\)

\(\Leftrightarrow 4x = 700\)

\(\Leftrightarrow x = 175\)(satisfies the conditions)

So the distance AB is 175 km long.

Average speed of motorbike: \(175 \div \frac{7}{2} = 50(km/h)\)

### 2.2. Exercise 2

Binh’s grandfather is older than Binh \(58\) years old. If the age of Binh’s father (or three) is added together and twice Binh’s age, it is equal to his age and the sum of the ages of all three is \(130\). Calculate Binh’s age.

**Solution guide**

Let \(x\) be Binh’s age (\(x\) positive integer).

Since Binh’s grandfather is \(58\) older than Binh, Binh’s age is \(x + 58\) (age)

Adding Binh’s father’s age and Binh’s age twice is equal to his age, so we have Binh’s father’s age:

\(\left( {x + 58} \right) – 2x \)\(=58 – x\) (age)

According to the assumption that the sum of the ages of three people equals \(130\), we have the equation:

\(\eqalign{ & x + \left( {x + 58} \right) + \left( {58 – x} \right) = 130 \cr&\Leftrightarrow x + x + 58 + 58 – x = 130 \cr & \Leftrightarrow x = 130 – 58 – 58 \cr} \)

\( \Leftrightarrow x = 14\) (satisfied)

So Binh \(14\) is old.

## 3. Practice

### 3.1. Essay exercises

**Question 1: **The first box contains \(60\) candy packs, the second box contains \(80\) candy packs. People took out three times as many candy packages from the second box as there were candies from the first box. How many packets of candy are removed from the first box, given that there are twice as many candy packs left in the first box as there are in the second box?

**Verse 2:** In a working session, class \(8A\) consisted of \(40\) students into two groups: the first group planted trees and the second group did the cleaning. The group that planted trees was larger than the group that did the cleaning, which was \(8\) people. How many students are there in the tree planting group?

**Question 3: **An odd two-digit number that is divisible by \(5\). The difference of that number and its tens digit is \(68\). Find that number.

**Question 4: **A fraction whose numerator is less than the denominator \(11.\) If the numerator is increased by \(3\) and the denominator is decreased by \(4\) then the fraction is equal to \(\displaystyle {3 \over 4}\). Find the original fraction.

**Question 5: **A car leaves Hanoi at \(\displaystyle8\) am, is expected to arrive in Hai Phong at \(\displaystyle10\) hours \(\displaystyle30\) minutes. But every hour the car was traveling slower than expected \(\displaystyle10 km\) so it took until \(\displaystyle11\) hours \(\displaystyle20\) minutes to arrive in Hai Phong. Calculate the distance from Hanoi to Hai Phong.

### 3.2. Multiple choice exercises

**Question 1: **Mom is 24 years older than me. After 2 years, the age of the mother will be 3 times the age of the son. Your child’s age is:

A. 5.

B. 10

C. 15

D. 20

**Verse 2:** Two consecutive even numbers whose product is 24 are:

A. 2;4

B. 4,6

C. 6,8

D. 8;10

A. 23.5cm

B. 47cm

C. 100cm

D. 3cm

A. 1h

B. 2h

C. 3h

D. 4h

A. 20km/h

B. 20km/h

C. 25km/h

D. 30km/h

## 4. Conclusion

Through this lesson, you will learn some of the main topics as follows:

- Understand how to solve problems by making equations
- How to choose a hide and its suitable condition

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