Class 7 Integers Word Problems | PDF - Free Printable

Let's solve the problems step by step.

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Question 1

Problem: In a school exam, Rohan’s score was up to 500 points. Then he scored -100 points in the words category. What was his score then?

Solution:
Rohan's initial score is 500 points. He then scores -100 points, which means he loses 100 points. To find his final score, we subtract 100 from 500:
\[
500 + (-100) = 500 - 100 = 400
\]

Answer: \(\boxed{400}\)

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Question 2

Problem: Kamla has gone for a hiking trip. She ended a hike at an elevation of 5,000 feet above sea level. She had started at an elevation of 5,000 feet above sea level. Which integer represents Kamla’s change in elevation?

Solution:
Kamla started her hike at an elevation of 5,000 feet and ended at the same elevation of 5,000 feet. Since there is no change in elevation, the change is:
\[
5,000 - 5,000 = 0
\]

Answer: \(\boxed{0}\)

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Question 4

Problem: Tell whether the following statements are always true, sometimes true, or always false.
a. If a positive integer is subtracted from a negative integer, the difference is a negative integer.
b. If a positive integer is subtracted from a positive integer, the difference is a positive integer.

Solution:

#### Part (a)
- Let the positive integer be \( p \) and the negative integer be \( -n \), where \( p > 0 \) and \( n > 0 \).
- The difference is:
\[
-n - p = -(n + p)
\]
Since \( n + p \) is a positive number, \(-(n + p)\) is a negative number.
- Therefore, the statement is always true.

#### Part (b)
- Let the two positive integers be \( a \) and \( b \), where \( a > 0 \) and \( b > 0 \).
- The difference is:
\[
a - b
\]
- If \( a > b \), then \( a - b \) is positive.
- If \( a = b \), then \( a - b = 0 \).
- If \( a < b \), then \( a - b \) is negative.
- Therefore, the statement is sometimes true (only when \( a > b \)).

Answers:
a. \(\boxed{\text{always true}}\)
b. \(\boxed{\text{sometimes true}}\)

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Question 5

Problem: Find solutions for the following problems:
a. Both numbers are less than 10. The distance between the two numbers on the number line is 14.
b. Both numbers are greater than -15 and less than 5. One number is 6 greater than the other number.
c. The numbers are opposites. The distance between the two numbers on the number line is 18.
d. One number is -35. The distance between the two numbers on the number line is 20.

Solution:

#### Part (a)
- Let the two numbers be \( x \) and \( y \). We know:
\[
|x - y| = 14 \quad \text{and} \quad x, y < 10
\]
- Since the distance is 14, one number must be 14 units away from the other. However, both numbers are less than 10. This is impossible because the maximum possible distance between two numbers less than 10 is less than 10.
- No solution exists.

Answer: \(\boxed{\text{No solution}}\)

#### Part (b)
- Let the two numbers be \( x \) and \( y \). We know:
\[
x = y + 6 \quad \text{and} \quad -15 < x, y < 5
\]
- Substitute \( x = y + 6 \) into the inequality:
\[
-15 < y + 6 < 5
\]
Subtract 6 from all parts:
\[
-15 - 6 < y < 5 - 6 \implies -21 < y < -1
\]
- Since \( y \) must be greater than -15 and less than 5, the valid range for \( y \) is:
\[
-15 < y < -1
\]
- Choose \( y = -2 \) (a value within the range):
\[
x = y + 6 = -2 + 6 = 4
\]
- Check the conditions:
- Both \( x = 4 \) and \( y = -2 \) are between -15 and 5.
- The distance is \( |4 - (-2)| = |4 + 2| = 6 \), which matches the condition \( x = y + 6 \).

Answer: \(\boxed{-2 \text{ and } 4}\)

#### Part (c)
- Let the two numbers be \( x \) and \( -x \) (since they are opposites). We know:
\[
|x - (-x)| = 18 \implies |x + x| = 18 \implies |2x| = 18
\]
- Solve for \( x \):
\[
2x = 18 \quad \text{or} \quad 2x = -18 \implies x = 9 \quad \text{or} \quad x = -9
\]
- The two numbers are \( 9 \) and \( -9 \).

Answer: \(\boxed{9 \text{ and } -9}\)

#### Part (d)
- Let the two numbers be \( x \) and \( -35 \). We know:
\[
|x - (-35)| = 20 \implies |x + 35| = 20
\]
- Solve for \( x \):
\[
x + 35 = 20 \quad \text{or} \quad x + 35 = -20
\]
\[
x = 20 - 35 = -15 \quad \text{or} \quad x = -20 - 35 = -55
\]
- The two numbers are \( -15 \) and \( -55 \).

Answer: \(\boxed{-15 \text{ and } -55}\)

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Final Answers:

1. \(\boxed{400}\)
2. \(\boxed{0}\)
4. a. \(\boxed{\text{always true}}\), b. \(\boxed{\text{sometimes true}}\)
5. a. \(\boxed{\text{No solution}}\), b. \(\boxed{-2 \text{ and } 4}\), c. \(\boxed{9 \text{ and } -9}\), d. \(\boxed{-15 \text{ and } -55}\)