One quarter of the sum of two consecutive odd numbers is 14.Find the odd numbers
n and n + 2
(1/4) (2 n+2) = 14
2 n + 2 = 56
2 n = 54
n = 27
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check
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(27+29) /4
56/4
14 sure enough
Odd numbers have the general form:
n = 2 k - 1
where
k = real number (1, 2, 3...)
The first number is:
n1 = 2 k - 1
The difference between two consecutive odd numbers is 2, so the second number is:
n2 = n1 + 2 = 2 k - 1 + 2 = 2 k + 1
One quarter of the sum of two consecutive odd numbers is 14 means:
1 / 4 ( n1 + n2 ) = 14
Multiply both sides by 4.
n1 + n2 = 56
2 k - 1 + 2 k + 1 = 56
4 k = 56
k = 56 / 4
k = 14
The first odd number is:
n1 = 2 k - 1
n1 = 2 • 14 - 1 = 28 - 1 = 27
The second odd number is:
n2 = 2 k + 1
n2 = 2 • 14 + 1 = 28 + 1 = 29
So two consecutive odd numbers are 27 and 29.
Check the result.
( 27 + 29 ) / 4 = 56 / 4 = 14
Let's start by assigning variables to the odd numbers. Let's call them x and x + 2.
According to the problem, one quarter of their sum is equal to 14.
The sum of the two numbers is x + (x + 2) = 2x + 2.
One quarter of their sum can be expressed as (1/4) * (2x + 2).
Setting this expression equal to 14, we have:
(1/4) * (2x + 2) = 14.
To solve for x, we can multiply both sides of the equation by 4 to get rid of the fraction:
2x + 2 = 56.
Next, subtract 2 from both sides:
2x = 54.
Finally, divide both sides by 2:
x = 27.
So, the first odd number is 27 and the next consecutive odd number is x + 2 = 27 + 2 = 29.
Therefore, the two odd numbers are 27 and 29.
Let's solve this step by step.
Step 1: Let's assume the two consecutive odd numbers as "n" and "n+2".
Note: Consecutive odd numbers are always 2 units apart. For example, 1 and 3 are consecutive odd numbers, 5 and 7 are consecutive odd numbers, and so on.
Step 2: The sum of two consecutive odd numbers is (n + n + 2).
Step 3: One-quarter of the sum is represented as (1/4)(n + n + 2).
Step 4: According to the problem, one-quarter of the sum is equal to 14, so we can write the equation as:
(1/4)(n + n + 2) = 14.
Step 5: Simplify the equation:
(1/4)(2n + 2) = 14.
Step 6: Multiply both sides by 4 to get rid of the fraction:
2n + 2 = 56.
Step 7: Subtract 2 from both sides:
2n = 54.
Step 8: Divide both sides by 2:
n = 27.
Step 9: Determine the consecutive odd numbers:
The first odd number is n = 27.
The second odd number is n + 2 = 27 + 2 = 29.
So, the two consecutive odd numbers are 27 and 29.