Which of the following numbers could be added to 3/14 to make a sum greater than 1/2?
one-fourth
one-third
two-sevenths
three-fourteenths
To determine which of the numbers could be added to 3/14 to make a sum greater than 1/2, we can convert 1/2 to an equivalent fraction with a denominator of 14 for easier comparison. To do this, we multiply both the numerator and the denominator by 7 (since 14 is 7 times 2):
1/2 = (1 * 7) / (2 * 7) = 7/14
Now we need to check which of the given numbers, when added to 3/14, results in a sum greater than 7/14.
Let's look at each option:
One-fourth (1/4):
To add 1/4 to 3/14, we first need to find a common denominator. The least common multiple (LCM) of 4 and 14 is 28. So we convert each fraction:
1/4 = (1 * 7) / (4 * 7) = 7/28
3/14 = (3 * 2) / (14 * 2) = 6/28
Adding them together gives us:
7/28 + 6/28 = 13/28
We need to compare this sum to 1/2 in the common denominator of 28:
1/2 = (1 * 14) / (2 * 14) = 14/28
Since 13/28 < 14/28, one-fourth is not enough to make the sum greater than 1/2.
One-third (1/3):
Now let's find a common denominator for 1/3 and 3/14. The LCM of 3 and 14 is 42.
1/3 = (1 * 14) / (3 * 14) = 14/42
3/14 = (3 * 3) / (14 * 3) = 9/42
Adding them together gives us:
14/42 + 9/42 = 23/42
Comparing this to 1/2 with a common denominator of 42:
1/2 = (1 * 21) / (2 * 21) = 21/42
Since 23/42 > 21/42, one-third is enough to make the sum greater than 1/2.
Two-sevenths (2/7):
To add 2/7 to 3/14, we can find a common denominator, which is 14 in this case.
2/7 = (2 * 2) / (7 * 2) = 4/14
3/14 = (3 * 1) / (14 * 1) = 3/14
Adding them gives us:
4/14 + 3/14 = 7/14
We already know that 7/14 is equivalent to 1/2, so two-sevenths added to 3/14 will exactly make 1/2, but it will not be greater than 1/2.
Three-fourteenths (3/14):
If we add another three-fourteenths to 3/14:
3/14 + 3/14 = 6/14
Converting 6/14 to have the same denominator as 1/2 with a denominator of 14:
6/14 is equivalent to 3/7, and we know that this is less than 7/14 or 1/2.
Therefore, the only fraction among the options provided that, when added to 3/14, makes a sum greater than 1/2, is one-third (1/3).