In the adult-school classes held several nights per week, 1/2 of the adults enrolled fall into the 41-60 age group, 1/3 of the remainder are over 60, 14 are in the 30-40 age group, and 12 are under 30. How many adults are enrolled in the school?
I did
1/2x + 1/6x + 14 + 12 = x
2/3x + 26 = x
26*3 = 78
Is 78 adults correct?
looks good to me
Well, my math skills might be a little rusty, but let me give it a shot!
So, you started with 1/2x representing the number of adults in the 41-60 age group. Then, you had 1/6x representing the number of adults who are over 60. Next, you added 14 for the 30-40 age group and 12 for those under 30.
Now, let's see if we can simplify that equation.
1/2x + 1/6x + 14 + 12 = x
Combining the fractions, we get:
3/6x + 1/6x + 14 + 12 = x
Simplifying further:
4/6x + 26 = x
Hmm, I'm afraid I'm not seeing how you arrived at 26*3 = 78. It seems like there may be an error somewhere. Let's try to solve the equation.
Subtracting 4/6x from both sides:
26 = 2/6x
Multiplying both sides by 6 to get rid of the fraction:
156 = 2x
Dividing both sides by 2:
78 = x
Ah, it looks like you were right after all! So, if my calculations are correct, there are 78 adults enrolled in the school. Well done! I'm relieved I didn't make a clown out of myself this time! Keep up the good work!
To solve this problem, let's break it down step by step.
Step 1: Let's assign variables to the unknowns.
Let x be the total number of adults enrolled in the school.
Step 2: Determine the number of adults in the 41-60 age group.
Half of the adults fall into the 41-60 age group, so the number of adults in this group is (1/2)*x.
Step 3: Determine the remaining number of adults after the 41-60 age group.
The remainder of the adults is (1 - 1/2)*x = (1/2)*x.
Step 4: Determine the number of adults over 60 years old.
One-third of the remaining adults are over 60 years old, so the number of adults in this group is (1/3)*[(1/2)*x] = (1/6)*x.
Step 5: Determine the number of adults in the 30-40 age group.
Given that there are 14 adults in the 30-40 age group, we have 14 = (1/6)*x.
Solving for x, we get:
x = 14 * (6/1) = 84.
Step 6: Check if the solution is correct.
Let's substitute the value of x into our original equation to verify if it satisfies the conditions:
(1/2)x + (1/6)x + 14 + 12 = x.
(1/2)(84) + (1/6)(84) + 14 + 12 = 84.
42 + 14 + 56 + 26 = 84.
126 + 40 = 84.
166 ≠ 84.
Step 7: Recap and conclude.
The original equation does not hold true when substituting x = 84, which means our initial solution is incorrect.
There is a mistake in your calculations. Let's try solving it again:
Step 1: Let's assign variables to the unknowns.
Let x be the total number of adults enrolled in the school.
Step 2: Determine the number of adults in the 41-60 age group.
Half of the adults fall into the 41-60 age group, so the number of adults in this group is (1/2)*x.
Step 3: Determine the remaining number of adults after the 41-60 age group.
The remainder of the adults is (1 - 1/2)*x = (1/2)*x.
Step 4: Determine the number of adults over 60 years old.
One-third of the remaining adults are over 60 years old, so the number of adults in this group is (1/3)*[(1/2)*x] = (1/6)*x.
Step 5: Determine the number of adults in the 30-40 age group.
Given that there are 14 adults in the 30-40 age group, we have 14 = (1/6)*x.
Solving for x, we get:
x = 14 * (6/1) = 84.
Step 6: Check if the solution is correct.
Let's substitute the value of x into our original equation to verify if it satisfies the conditions:
(1/2)x + (1/6)x + 14 + 12 = x.
(1/2)(84) + (1/6)(84) + 14 + 12 = 84.
42 + 14 + 56 + 26 = 84.
126 + 40 = 84.
166 ≠ 84.
Step 7: Recap and conclude.
The original equation does not hold true when substituting x = 84, which means our initial solution is incorrect.
Therefore, there is an error in your calculations.
To solve this problem correctly, let's set up the equation again:
(1/2)x + (1/6)x + 14 + 12 = x
Combine like terms:
(3/6)x + (1/6)x + 26 = x
Simplify:
(4/6)x + 26 = x
Multiply both sides by 6 to get rid of the fractions:
4x + 156 = 6x
Subtract 4x from both sides:
156 = 2x
Divide both sides by 2:
x = 78
So, the correct answer is 78 adults enrolled in the school.
To solve this problem, you correctly set up an equation based on the given information. Let me walk you through the steps:
1. Start by letting x represent the total number of adults enrolled in the school.
2. From the given information, we know that 1/2 of the adults fall into the 41-60 age group. So, the number of adults in this age group is (1/2)x.
3. The remainder is the leftover adults after the 41-60 age group, which is (x - (1/2)x) = (1/2)x.
4. Now, 1/3 of the remainder are over 60, which means the number of adults over 60 is (1/3) * (1/2)x = (1/6)x.
5. Additionally, 14 adults are in the 30-40 age group, and 12 adults are under 30. So, the total number of adults from these two age groups is 14 + 12 = 26.
6. Combining all the information, we can write the equation as: (1/2)x + (1/6)x + 26 = x.
Now, let's simplify and solve the equation:
1/2x + 1/6x + 26 = x
Common denominator for 1/2 and 1/6 is 6, so we get:
(3/6)x + (1/6)x + 26 = x
Adding the fractions, we have:
(4/6)x + 26 = x
Multiplying both sides by 6 to eliminate the denominator, we get:
4x + 156 = 6x
Subtracting 4x from both sides, we get:
156 = 2x
Dividing both sides by 2, we find:
x = 78
So, you are correct! There are 78 adults enrolled in the school.