**Worksheet on rational numbers :**

Rational numbers worksheet is much useful to the students who would like to practice problems on adding, subtracting, multiplying and dividing rational numbers.

1. Add : (-2/3) + (-1/3)

2. Add : (+2/5) + (+1/5)

3. Add : (-4/5) + (+7/5)

4. Add : (+5/9) + (-8/9)

5. Subtract "2.5" from "3.5" using a number line.

6. Subtract "1.5" from "4.5" using a number line.

7. Subtract -1.5 from 2.5.

8. Multiply -16 and -1/8.

9. Multiply 2 and -1/4.

10. Divide 2/3 by -7/6

11. Divide 9/5 by 3

12. Malachi hikes for 2.5 miles and stops for lunch. Then he hikes for 1.5 more miles. How many miles did he hike altogether?

13. During the day, the temperature increases by 4.5 degrees. At night, the temperature decreases by 7.5 degrees. What is the overall change in temperature?

14. During the hottest week of the summer, the water level of the Muskrat River was 5/6 foot below normal. The following week, the level was 1/3 foot below normal. What is the overall change in the water level ?

15. Gina hiked down a canyon and stopped each time she descended 1/2 mile to rest. She hiked a total of 4 sections. What is her overall change in elevation ?

16. A diver needs to descend to a depth of 100 feet below sea level. She wants to do it in 5 equal descents. How far should she travel in each descent ?

**Problem 1 : **

Add : (-2/3) + (-1/3)

**Solution : **

**Step 1 : **

Find the absolute values.

|-2/3| = 2/3 and |-1/3| = 1/3

**Step 2 : **

Find the sum of the absolute values :

2/3 + 1/3 = (2+1) / 3

2/3 + 1/3 = 3/3

2/3 + 1/3 = 1

**Step 3 : **

Use the sign of the rational numbers to write the sum.

(-2/3) + (-1/3) = -1

**Problem 2 : **

Add : (+2/5) + (+1/5)

**Solution : **

**Step 1 : **

Find the absolute values.

|+2/5| = 2/5 and |+1/5| = 1/5

**Step 2 : **

Find the sum of the absolute values :

2/5 + 1/5 = (2+1)/5

2/5 + 1/5 = 3/5

**Step 3 : **

Use the sign of the rational numbers to write the sum.

(+2/5) + (+1/5) = +3/5

**Problem 3 : **

Add : (-4/5) + (+7/5)

**Solution : **

**Step 1 : **

Find the absolute difference of the rational numbers without the actual signs.

|4/5 - 7/5| = |-3/5| = 3/5

**Step 2 : **

In the given two rational numbers 4/5 and 7/5, the sign of the bigger number is positive. So, we have to take positive sign to the answer.

Hence, (-4/5) + (+7/5) = +3/5

**Problem 4 : **

Add : (+5/9) + (-8/9)

**Solution : **

**Step 1 : **

Find the absolute difference of the rational numbers without the actual signs.

|5/9 - 8/9| = |-3/9| = 3/9 = 1/3

**Step 2 : **

In the given rational numbers 5 and 8, the sign of the bigger number is negative. So, we have to take negative sign to the answer.

Hence, (+5/9) + (-8/9) = -1/3

**Problem 5 : **

Subtract "2.5" from "3.5" using a number line.

**Solution : **

Since we subtract a positive rational number 2.5 from 3.5, we have to move 2.5 units in the negative direction from 3.5 on the number line as given in the picture below.

After having moved 2.5 units in the negative direction, we are in the position of "1"

Hence, 3.5 - (+2.5) = 1

**Problem 6 : **

Subtract "1.5" from "4.5" using a number line.

**Solution : **

Since we subtract a positive rational number 1.5 from 4.5, we have to move 1.5 units in the negative direction from 4.5 on the number line as given in the picture below.

After having moved 1.5 units in the negative direction, we are in the position of "3"

Hence, 4.5 - (+1.5) = 3

**Problem 7 : **

Subtract -1.5 from 2.5.

**Solution : **

**Step 1 : **

Since we subtract the negative rational number -1.5 from 2.5, we have to move 1.5 units from 2.5 in the positive direction on the number line.

It has been illustrated in the picture given below.

**Step 2 :**

After having moved 1.5 units in the positive direction from 2.5, we are in the position of "4"

Hence, 2.5 - (-1.5) = 4

**Problem 8 : **

Multiply -16 and -1/8.

**Solution : **

**Step 1 : **

In the two rational numbers -16 and -1/8, the signs are same.

**Step 2 : **

Find the product of 16 and 1/8

16 x 1/8 = 16/8 = 16/8 = 2

**Step 3 : **

Since we multiply two rational numbers with the same sign, the result is always positive.

Hence, -16 x (-1/8) = 2.

**Problem 9 : **

Multiply 2 and -1/4.

**Solution : **

**Step 1 : **

In the two rational numbers 2 and -1/4, the signs are different.

**Step 2 : **

Find the product of 2 and 1/4

2 x 1/4 = 2/4 = 1/2

**Step 3 : **

Since we multiply two rational numbers with different signs, the result is always negative.

Hence, 2 x (-1/4) = - 1/2.

Let us look at the next problem on "Worksheet on rational numbers".

**Problem 10 : **

Divide 2/3 by -7/6

**Solution :**

**Step 1 :**

Take the reciprocal of the second rational number.

-7/6 ---- reciprocal ----> -6/7

**Step 2 : **

Multiply the first rational number 2/3 by -6/7

(2/3) x (-6/7)

**Step 3 : **Simplify

(2/1) x (-2/7)

**Step 4 : **Multiply

(2/1) x (-2/7) = -4/7 Positive times negative equals negative

Hence, 2/3 ÷ -7/6 = -4/7

Let us look at the next problem on "Worksheet on rational numbers".

**Problem 11 : **

Divide 9/5 by 3

**Solution :**

**Step 1 :**

Take the reciprocal of the second rational number.

3 ---- reciprocal ----> 1/3

**Step 2 : **

Multiply the first rational number 9/5 by 1/3

(9/5) x (1/3)

**Step 3 : **Simplify

(3/5) x (1/1)

**Step 4 : **Multiply

(3/5) x (1/1) = 3/5 Positive times positive equals positive

Hence, 9/5 ÷ 3 = 3/5

Let us look at the next problem on "Worksheet on rational numbers".

**Problem 12 : **

Malachi hikes for 2.5 miles and stops for lunch. Then he hikes for 1.5 more miles. How many miles did he hike altogether?

**Solution : **

**Step 1 : **

Use positive numbers to represent the distance Malachi hiked.

**Step 2 : **

Find 2.5 + 1.5.

Let us use the real number line to add 2.5 and 1.5.

**Step 3 : **

Start at 2.5.

**Step 4 : **

Move 1.5 units to the right because the second addend is positive.

The result is 4.

Malachi hiked 4 miles.

Let us look at the next problem on "Worksheet on rational numbers".

**Problem 13 :**

During the day, the temperature increases by 4.5 degrees. At night, the temperature decreases by 7.5 degrees. What is the overall change in temperature?

**Solution :**

**Step 1 :**

Use a positive number to represent the increase in temperature and a negative number to represent a decrease in temperature.

**Step 2 : **

Find 4.5 + (-7.5).

Let us use the real number line to add 4.5 and (-7.5).

**Step 3 : **

Start at 4.5.

**Step 4 :**

Move | -7.5 | = 7.5 units to the left because the second addend is negative.

The result is -3.

The temperature decreased by 3 degrees overall.

Let us look at the next problem on "Worksheet on rational numbers".

**Problem 14 : **

During the hottest week of the summer, the water level of the Muskrat River was 5/6 foot below normal. The following week, the level was 1/3 foot below normal. What is the overall change in the water level ?

**Solution :**

Subtract to find the difference in water levels. That is, we have to find -1/3 - (-5/6)

**Step 1 : **

Since we subtract the negative rational number -5/6 from -1/3, we have to move 5/6 units from -1/3 in the positive direction on the number line.

It has been illustrated in the picture given below.

**Step 2 :**

After having moved 5/6 units in the positive direction from -1/3, we are in the position of "1/2"

Hence, the water level changed 1/2 foot.

Let us look at the next problem on "Worksheet on rational numbers".

**Problem 15 : **

Gina hiked down a canyon and stopped each time she descended 1/2 mile to rest. She hiked a total of 4 sections. What is her overall change in elevation?

**Solution : **

**Step 1 : **

Use a negative number to represent the change in elevation.

**Step 2 : **

Find 4 x (-1/2).

**Step 3 : **

Start at 0. Move 1/2 unit to the left 4 times.

The result is -2.

Hence, the overall change is -2 miles.

**Check : **

Use the rules for multiplying rational numbers.

4 x (-1/2) = - 4/2 A negative times a positive equals a the the the the te kdjhnegative.

4 x (-1/2) = - 2 Simplify

Let us look at the next problem on "Worksheet on rational numbers".

**Problem 16 : **

A diver needs to descend to a depth of 100 feet below sea level. She wants to do it in 5 equal descents. How far should she travel in each descent ?

**Solution :**

To find how far she should travel in each descent, we have to divide 100 by 5.

**Step 1 :**

Take the reciprocal of the divisor 5.

5 ---- reciprocal ----> 1/5

**Step 2 : **

Multiply 100 by 1/5

(100) x (1/5)

**Step 3 : **Simplify

(20) x (1/1)

**Step 4 : **Multiply

(20) x (1/1) = 20

Hence, she should travel 20 feet in each descent.

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