Half of a number added to a fifth of three less than the number is equal to two-thirds the number. What is the number?
my answer: the number is 18
x/2 + (x-3)/5 = 2/3x
x/30 + (6x-18)/30 = 20/30x
15x + 6x - 18 = 20x
18 = x
That's my answer too.
To solve this problem, let's break it down into steps:
Let's assume the number is 'x'.
Step 1: "Half of a number"
Half of a number is x/2.
Step 2: "A fifth of three less than the number"
Three less than the number would be x - 3. A fifth of that would be (x - 3)/5.
Step 3: "Half of a number added to a fifth of three less than the number"
Adding the two expressions from Step 1 and Step 2, we get x/2 + (x - 3)/5.
Step 4: "Two-thirds the number"
Two-thirds of the number is 2x/3.
Step 5: Set up the equation
According to the problem, the expression from Step 3 is equal to the expression from Step 4. So, we have the equation:
x/2 + (x - 3)/5 = 2x/3.
Now we can solve for x.
Step 6: Solve for x
Let's get rid of the denominators by multiplying each term by the least common multiple (LCM) of the denominators, which in this case is 30:
30(x/2) + 30((x - 3)/5) = 30(2x/3).
15x + 6(x - 3) = 20x.
15x + 6x - 18 = 20x.
21x - 18 = 20x.
Subtracting 20x from both sides, we get:
21x - 20x - 18 = 0.
x - 18 = 0.
Adding 18 to both sides, we finally get:
x = 18.
Therefore, the number is indeed 18.
To solve this problem algebraically, let's assign a variable to the unknown number. Let's call the number "x."
According to the problem, half of the number (x/2) added to a fifth of three less than the number (1/5 * (x - 3)) is equal to two-thirds of the number (2/3 * x).
Putting this all together, we can form the equation:
(x/2) + (1/5 * (x - 3)) = (2/3 * x)
Now, let's solve the equation step by step:
To get rid of the fractions, we can multiply the entire equation by the least common denominator (LCD), which is 30 in this case.
30 * ((x/2) + (1/5 * (x - 3))) = 30 * (2/3 * x)
15x + 6(x - 3) = 20x
Now, distribute and simplify:
15x + 6x - 18 = 20x
21x - 18 = 20x
To isolate the variable, we subtract 20x from both sides:
21x - 20x - 18 = 20x - 20x
x - 18 = 0
Next, add 18 to both sides to isolate x:
x - 18 + 18 = 0 + 18
x = 18
Therefore, the number x is indeed 18.