The lengths in feet of three ropes are consecutive multiples of three. If each rope were four feet shorter, the sum of their lengths would be 42 feet. Find the lengths of the three ropes.
let the length be x-3, x, and x+3
had each been 4 ft shorter, they would have been:
x-7, x-4, and x-1
x-7 + x-4 + x-1 = 42
3x = 54
x = 18
the ropes are 15, 18 and 21 ft each
Let's assume the length of the first rope is x feet.
According to the given information, the lengths of the three ropes are consecutive multiples of three, so the length of the second rope would be (x + 3) feet, and the length of the third rope would be (x + 6) feet.
If each rope were four feet shorter, the new lengths would be:
First rope: (x - 4) feet
Second rope: (x + 3 - 4) feet = (x - 1) feet
Third rope: (x + 6 - 4) feet = (x + 2) feet
Now we can form the equation based on the given information:
(x - 4) + (x - 1) + (x + 2) = 42
Simplifying the equation:
3x - 3 = 42
3x = 42 + 3
3x = 45
x = 45 / 3
x = 15
So, the length of the first rope is 15 feet.
The length of the second rope is (15 + 3) = 18 feet.
The length of the third rope is (15 + 6) = 21 feet.
Therefore, the lengths of the three ropes are 15 feet, 18 feet, and 21 feet.
To solve this problem, let's break it down step by step.
Step 1: Understand the problem
We are given three ropes, and the lengths of these ropes are consecutive multiples of three. Let's call the lengths of the ropes x, y, and z. We know that x, y, and z are consecutive multiples of three, so we can express them as:
x = 3k
y = 3(k+1)
z = 3(k+2)
where k is an integer.
Step 2: Identify the given information
We are told that if each rope were four feet shorter, the sum of their lengths would be 42 feet. From this information, we can form an equation.
Step 3: Formulate an equation
If we subtract 4 feet from each rope, we can write the equation as follows:
(x-4) + (y-4) + (z-4) = 42
Step 4: Simplify the equation
Substituting the expressions for x, y, and z from step 1 into the equation, we have:
(3k-4) + (3(k+1)-4) + (3(k+2)-4) = 42
Simplifying further, we get:
3k-4 + 3k+3 - 4 + 3k+6 - 4 = 42
9k + 1 = 42
9k = 42 - 1
9k = 41
k = 41/9
k ≈ 4.56
Step 5: Find the lengths of the three ropes
Since k is not an integer, we need to find the nearest integer value for k that makes the lengths of the ropes integers. The nearest integer value of k is 5.
Substituting k = 5 back into the expressions for x, y, and z from step 1, we find:
x = 3(5) = 15 feet
y = 3(5+1) = 18 feet
z = 3(5+2) = 21 feet
Therefore, the lengths of the three ropes are 15 feet, 18 feet, and 21 feet.