College

6. Abdu is 10 years older than Hailu. Three years ago, Abdu was 3 times as old as Hailu. Find their current ages.

7. The sum of two consecutive integers is three times their difference. What is the larger number?

8. The sum of eight times a number and five times another number is 184. The difference of the second number from the first is -3. Find the numbers.

9. The perimeter of a rectangular field is 628 meters. The length of the field exceeds its width by 6 meters. Find the dimensions of the rectangle.

10. Suppose the current temperature is 54°F. It is expected to rise 2°F each hour for the next several hours. In how many hours will the temperature be 78°F?

Answer :

Sure! Let's go through each problem step-by-step.

### Problem 6:

Question: Abdu is 10 years older than Hailu. Three years ago, Abdu was three times as old as Hailu. Find their ages.

1. Let's say Hailu's current age is [tex]\( h \)[/tex].
2. Therefore, Abdu's current age would be [tex]\( h + 10 \)[/tex].
3. Three years ago, Hailu's age was [tex]\( h - 3 \)[/tex] and Abdu's age was [tex]\( h + 10 - 3 = h + 7 \)[/tex].
4. According to the problem, three years ago Abdu was three times as old as Hailu. So, the equation becomes:
[tex]\[
h + 7 = 3(h - 3)
\][/tex]
5. Solving the equation, we find that:
[tex]\[
h + 7 = 3h - 9
\][/tex]
[tex]\[
16 = 2h
\][/tex]
[tex]\[
h = 8
\][/tex]
6. Therefore, Hailu is 8 years old.
7. Abdu's age is [tex]\( h + 10 = 8 + 10 = 18 \)[/tex].

Answer: Hailu is 8 years old and Abdu is 18 years old.

### Problem 7:

Question: The sum of two consecutive integers is three times their difference. What is the larger number?

1. Let the first integer be [tex]\( n \)[/tex]. So, the next consecutive integer is [tex]\( n + 1 \)[/tex].
2. The sum of these numbers is [tex]\( n + (n + 1) = 2n + 1 \)[/tex].
3. The difference between these numbers is [tex]\( (n + 1) - n = 1 \)[/tex].
4. According to the problem, the sum is three times their difference:
[tex]\[
2n + 1 = 3 \cdot 1
\][/tex]
[tex]\[
2n + 1 = 3
\][/tex]
[tex]\[
2n = 2
\][/tex]
[tex]\[
n = 1
\][/tex]
5. Therefore, the numbers are 1 and 2, with 2 being the larger number.

Answer: The larger number is 2.

### Problem 8:

Question: The sum of eight times a number and five times another number is 184. The difference of the second number from the first is -3. Find the numbers.

1. Let the first number be [tex]\( x \)[/tex] and the second number be [tex]\( y \)[/tex].
2. We have two equations:
[tex]\[
8x + 5y = 184
\][/tex]
[tex]\[
x - y = -3
\][/tex]
3. Solve the second equation for [tex]\( x \)[/tex]:
[tex]\[
x = y - 3
\][/tex]
4. Substitute [tex]\( x = y - 3 \)[/tex] into the first equation:
[tex]\[
8(y - 3) + 5y = 184
\][/tex]
[tex]\[
8y - 24 + 5y = 184
\][/tex]
[tex]\[
13y = 208
\][/tex]
[tex]\[
y = 16
\][/tex]
5. Using [tex]\( y = 16 \)[/tex], find [tex]\( x \)[/tex]:
[tex]\[
x = 16 - 3 = 13
\][/tex]

Answer: The numbers are 13 and 16.

### Problem 9:

Question: The perimeter of a rectangular field is 628 meters. The length of the field exceeds its width by 6 meters. Find the dimensions of the rectangle.

1. Let the width be [tex]\( w \)[/tex]. Then the length is [tex]\( w + 6 \)[/tex].
2. The perimeter of a rectangle is calculated as [tex]\( 2 \times (\text{Length} + \text{Width}) \)[/tex].
3. Set up the equation using the given perimeter:
[tex]\[
2(w + (w + 6)) = 628
\][/tex]
[tex]\[
2(2w + 6) = 628
\][/tex]
[tex]\[
4w + 12 = 628
\][/tex]
[tex]\[
4w = 616
\][/tex]
[tex]\[
w = 154
\][/tex]
4. Therefore, the width is 154 meters, and the length is [tex]\( 154 + 6 = 160 \)[/tex] meters.

Answer: The dimensions are 154 meters by 160 meters.

### Problem 10:

Question: Suppose the current temperature is 54°F. It is expected to rise 2°F each hour for the next several hours. In how many hours will the temperature be 78°F?

1. Let the number of hours be [tex]\( h \)[/tex].
2. The temperature after [tex]\( h \)[/tex] hours will be:
[tex]\[
54 + 2h = 78
\][/tex]
3. Solving for [tex]\( h \)[/tex]:
[tex]\[
2h = 78 - 54
\][/tex]
[tex]\[
2h = 24
\][/tex]
[tex]\[
h = 12
\][/tex]

Answer: It will take 12 hours for the temperature to reach 78°F.