one sixth of the sum of twice a number and 1 is the same as one quarter of the difference between three times the number and 1. find the number?
1/6(2x+1)=1/4(3x-1)
(2x+1)/6 = (3x-1)/4
Let's break down the problem step by step:
Step 1: Let's represent the unknown number with "x."
Step 2: The sum of twice a number (2x) and 1 is given as (2x + 1).
Step 3: One-sixth of this sum is (1/6) * (2x + 1).
Step 4: The difference between three times the number (3x) and 1 is given as (3x - 1).
Step 5: One-quarter of this difference is (1/4) * (3x - 1).
Step 6: According to the problem statement, the above two expressions are equal:
(1/6) * (2x + 1) = (1/4) * (3x - 1)
Step 7: We can now solve the equation to find the value of x.
Multiplying both sides of the equation by 12 to eliminate fractions:
12 * (1/6) * (2x + 1) = 12 * (1/4) * (3x - 1)
2(2x + 1) = 3(3x - 1)
4x + 2 = 9x - 3
Moving the variables terms to one side of the equation and constants to the other side:
4x - 9x = -3 - 2
-5x = -5
Dividing both sides of the equation by -5:
x = -5/-5
x = 1
Therefore, the number is 1.
To find the number, let's break down the problem step by step.
Let's assume the number we are looking for is represented by the variable x.
Step 1: Translate the problem into an equation:
"One sixth of the sum of twice a number and 1 is the same as one quarter of the difference between three times the number and 1."
The equation for the given problem can be written as:
(1/6)(2x + 1) = (1/4)(3x - 1)
Step 2: Multiply both sides of the equation by the least common multiple (LCM) of 6 and 4 to eliminate the fractions. The LCM of 6 and 4 is 12.
12 * (1/6)(2x + 1) = 12 * (1/4)(3x - 1)
Simplifying, we get:
2(2x + 1) = 3(3x - 1)
Step 3: Distribute and solve for x.
4x + 2 = 9x - 3
Subtract 4x from both sides:
2 = 5x - 3
Add 3 to both sides:
5 = 5x
Divide both sides by 5:
x = 1
Therefore, the number we are looking for is x = 1.