### Key Concepts

- Decide which operations to use to solve problems
- Use properties of operations with rational numbers
- Solve multi-step problems with rational numbers

## Introduction:

- Solve problems using basic operations
- Identify the operations to be used to solve the real-life context problems
- Solving problems using distributive property
- Solving multi–step problems with rational numbers

## 1.10.1 Decide which operations to use to solve problems

**Example 1:**

A truck is traveling at a speed of 7 miles per hour. How long will it take the truck to travel 4½ miles.

**Solution:**

Decide which operation to use to find the truck’s distance.

7/ 4 ½

= 7 ÷ 4½ = 7 ÷ 9/2

= 7 × 2/9 = 7×2 / 9 = 14/9 miles.

So, the truck will take = 14 / 9 miles to travel.

**Example 2:**

Thomas has a 7(3/2) inch long board. If he cuts it into 9 equal pieces, then what would be the length of each piece?

**Solution:**

Total length of the board = 7 ^{3/2}

Total number of pieces = 9

Using operations to find the length of each piece, we get

9 / 7 ^{3/2}

= 9 ÷ 7 ^{3/2} = 9 ÷ 17 / 2

= 9 × 2/ 17 = 9×2 / 17 = 18 / 17

So, the length of each piece = 18/ 17 inch.

### 1.10.2 Use properties of operations with rational numbers

**Example 3:**

Brianna played a trivia game. Total number of questions are 15, and for each correct answer, she will gain 2 points, and for each incorrect answer, she will lose point. What is the total score of Brianna?

**Solution:**

Here, we can solve the question using rational number properties of operations.

**Method 1:**

(15)2 ¼+ (15)(-1/2)

=9/4(15) +(-1/2)(15)

=135 / 4 + (-15/2)

= 135−30 / 4 =105 / 4= 26 ¼

Brianna’s score is 26 ¼ points.

**Method 2:**

(15)2 ¼+ (15)(-1/2)

= 15[2 ¼ + (-1/2)] (Using distributive property)

= 15(9/4 – ½) = 15(1 ¾)

= 15( 7/4 ) = 105 / 4 = 26 ¼

Brianna’s score is 26 ¼ points.

**Example 4:**

Jecinda attempted an online quiz. Total number of questions are 20, and for each correct answer, she will gain 2 points, and for each incorrect answer, she will lose point. If she answers 10 correct answers and 10 incorrect answers. What is Jecinda’s total score?

**Solution:**

(10)2 ¼+ (15)(-1/2)

= 10[2 ¼ + (-1/2)] (Using distributive property)

= 10(9/4 – ½) = 10(1 ¾)

= 10( 7/4 ) = 47 / 4 = 11 ¾

Jecinda’s score is 11 ¾ points.

### 1.10.3 Solve multi-step problems with rational numbers

**Example 5:**

The temperature at 11:00 AM was and increased each hour for the next 3 hours. Find the temperature at 2:00 PM.

**Solution:**

**Step 1:**

Multiply to find the total change in temperature.

1.5 × 3 = 4.5

The total change in the temperature is 4.5 degrees.

**Step 2:**

Add the total change in the temperature to the original temperature.

4.5 + ( 4) = + 0.5 The temperature at 2:00 PM is

## Exercise:

- At 37.5 feet, the boat drops an anchor deep into the river. If the anchor falls at a rate of 0.8 feet per second. Calculate the total time taken to reach the anchor deep into the river?
- A scuba diver dives 32 feet in 12 seconds to the bottom of the river. Find the change in the diver’s position per second.
- Magda has 85 hair barrettes. Peter has 43 hair barrettes. What is the total number of hair barrettes?
- Bob has a 27 inch box. If he cuts it into 9 equal-sized pieces, what is the measure of each piece?
- Seven equal-sized boxes weigh 30 pounds. What is the weight of each box?
- Bonnie has 30 dollars. If she splits it into 5 equal groups, how many dollars will each group have?
- The cost of meters of sheet is . Find the total cost of one meter sheet.
- Makram earns $12000 per month. He spends of his income on food; of his income on house rent and of the remainder on the education of children. How much money is still left with him?
- A car is moving at a speed of miles per hour. What will be the distance covered in hours?
- The temperature at 12:00 PM is 10 . It drops 1 each hour for the next 5 hours. What was the temperature at 5:00 PM?

### What have we learnt:

- Solve problems using basic operations.
- Identify the operations to be used to solve the real-life context problems.
- Solving problems using distributive property.
- Solving multi-step problems with rational numbers.

### Concept Map

#### Related topics

#### Addition and Multiplication Using Counters & Bar-Diagrams

Introduction: We can find the solution to the word problem by solving it. Here, in this topic, we can use 3 methods to find the solution. 1. Add using counters 2. Use factors to get the product 3. Write equations to find the unknown. Addition Equation: 8+8+8 =? Multiplication equation: 3×8=? Example 1: Andrew has […]

Read More >>#### Dilation: Definitions, Characteristics, and Similarities

Understanding Dilation A dilation is a transformation that produces an image that is of the same shape and different sizes. Dilation that creates a larger image is called enlargement. Describing Dilation Dilation of Scale Factor 2 The following figure undergoes a dilation with a scale factor of 2 giving an image A’ (2, 4), B’ […]

Read More >>#### How to Write and Interpret Numerical Expressions?

Write numerical expressions What is the Meaning of Numerical Expression? A numerical expression is a combination of numbers and integers using basic operations such as addition, subtraction, multiplication, or division. The word PEMDAS stands for: P → Parentheses E → Exponents M → Multiplication D → Division A → Addition S → Subtraction Some examples […]

Read More >>#### System of Linear Inequalities and Equations

Introduction: Systems of Linear Inequalities: A system of linear inequalities is a set of two or more linear inequalities in the same variables. The following example illustrates this, y < x + 2…………..Inequality 1 y ≥ 2x − 1…………Inequality 2 Solution of a System of Linear Inequalities: A solution of a system of linear inequalities […]

Read More >>
Comments: