The sum of two consecutive even numbers is 278. What are the two numbers?
A 50m cable is cut into three sections. Two sections are the same length and the third section is 2m longer than the other two. Find the length of each section?
n + (n + 2) = 278 ... 2 n + 2 = 278
2 s + (s + 2) = 50 ... 3 s + 2 = 50
To find the two consecutive even numbers whose sum is 278, we can let the first even number be x. Since they are consecutive even numbers, the second even number would be x + 2.
So, the equation to represent the sum of the two consecutive even numbers is:
x + (x + 2) = 278
To solve this equation, we will combine like terms:
2x + 2 = 278
Next, we will isolate the variable x by subtracting 2 from both sides of the equation:
2x = 276
Finally, we will divide both sides by 2 to solve for x:
x = 138
Thus, the first even number is 138, and the second even number is:
138 + 2 = 140
Therefore, the two numbers are 138 and 140.
Now, let's move on to the second question.
Let's represent the length of the two equal sections as x. The length of the third section, which is 2m longer, would be x + 2.
Since the cable is cut into three sections, the sum of their lengths would be equal to the length of the original cable, which is 50m.
So, the equation to represent this would be:
x + x + (x + 2) = 50
To solve this equation, we will combine like terms:
3x + 2 = 50
Next, we will isolate the variable x by subtracting 2 from both sides:
3x = 48
Finally, we will divide both sides by 3 to solve for x:
x = 16
Thus, the length of the equal sections is 16m, and the length of the third section is:
16 + 2 = 18m
Therefore, the lengths of the three sections are 16m, 16m, and 18m.
To solve the first question, which involves finding two consecutive even numbers with a given sum, we can use algebra.
Let's assume the first even number is "x". Since it is even, we know it can be written as 2n, where "n" is a positive integer. So, the second consecutive even number can be represented as "x + 2".
According to the given information, the sum of these two consecutive even numbers is 278, so we can write an equation:
x + x + 2 = 278
Simplifying the equation:
2x + 2 = 278
Subtracting 2 from both sides:
2x = 276
Dividing both sides by 2:
x = 138
So, the first even number is 138. To find the second even number, we can substitute this value into the equation:
138 + 2 = 140
Therefore, the two consecutive even numbers whose sum is 278 are 138 and 140.
Now, moving on to the second question, let's assume the length of one of the equal sections is "x". Since the second section is the same length, we can also denote it as "x". And since the third section is 2m longer, we can denote it as "x + 2".
According to the given information, the total length of the cable is 50m, so we can write an equation:
x + x + (x + 2) = 50
Simplifying the equation:
3x + 2 = 50
Subtracting 2 from both sides:
3x = 48
Dividing both sides by 3:
x = 16
So, the length of one of the equal sections is 16m. Substituting this value into the equation to find the length of the other equal section:
16 + 16 = 32
Therefore, the length of each of the two equal cable sections is 16m, and the length of the third section is 32m.