Easy Method of Solving Problems with Rational Numbers

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:

1. 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?
2. 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.
3. Magda has 85 hair barrettes. Peter has 43 hair barrettes. What is the total number of hair barrettes?
4. Bob has a 27  inch box. If he cuts it into 9 equal-sized pieces, what is the measure of each piece?
5. Seven equal-sized boxes weigh 30 pounds. What is the weight of each box?
6. Bonnie has 30 dollars. If she splits it into 5 equal groups, how many dollars will each group have?
7. The cost of  meters of sheet is . Find the total cost of one meter sheet.
8. 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?
9. A car is moving at a speed of  miles per hour. What will be the distance covered in  hours?
10. 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