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.
### 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.
