A ant walks all the way around a triangle drawn in the dirt. Two of the dimensions of the triangle are 5 1/10 centimeters and 7 1/5 centimeters. The total distance traveled by the ant is 20 centimeters. What is the length of the third side of the triangle?
Let the length of the third side of the triangle be $x$ centimeters. Then $5 \frac{1}{10} + 7 \frac{1}{5} + x = 20.$ Converting all the fractions to twentieths, we get
\[5 \frac{2}{20} + 7 \frac{4}{20} + x = 20.\]We can rewrite this as
\[\frac{104}{20} + x = 20.\]Then $x = 20 - \frac{104}{20} = \frac{20 \cdot 20 - 104}{20} = \frac{296}{20} = \boxed{14.8}$ centimeters.
I don't understand what you said
Let's start by labeling the sides of the triangle. Let $AB$ be the side with length $5 \frac{1}{10}$ centimeters and $AC$ be the side with length $7 \frac{1}{5}$ centimeters. We want to find the length of the third side, so let's call it $BC$, with length $x$ centimeters.
The total distance traveled by the ant is the sum of the lengths of all three sides of the triangle, which we know is $20$ centimeters. So we can write the equation:
\[AB + AC + BC = 20.\]
Substituting in the lengths of $AB$ and $AC$, we have:
\[5 \frac{1}{10} + 7 \frac{1}{5} + x = 20.\]
Next, let's convert the mixed fractions to improper fractions. We get:
\[\frac{51}{10} + \frac{36}{5} + x = 20.\]
To solve for $x$, we first add the two fractions:
\[\frac{51}{10} + \frac{36}{5} = \frac{51 \cdot 2}{10 \cdot 2} + \frac{36}{5} = \frac{102}{20} + \frac{36}{5}.\]
Now, we write $20$ as $\frac{20}{1}$ to have the same denominator:
\[\frac{102}{20} + \frac{36}{5} = \frac{102}{20} + \frac{36}{5} \cdot \frac{4}{4} = \frac{102}{20} + \frac{144}{20}.\]
Adding the fractions, we get:
\[\frac{102}{20} + \frac{144}{20} = \frac{102 + 144}{20} = \frac{246}{20}.\]
So, our equation becomes:
\[\frac{246}{20} + x = 20.\]
To solve for $x$, we isolate $x$ by subtracting $\frac{246}{20}$ from both sides:
\[x = 20 - \frac{246}{20}.\]
To simplify the right side, we write $20$ as $\frac{20}{1}$ to have the same denominator:
\[x = \frac{20}{1} - \frac{246}{20} = \frac{20}{1} \cdot \frac{20}{20} - \frac{246}{20} = \frac{400}{20} - \frac{246}{20} = \frac{400 - 246}{20} = \frac{154}{20}.\]
Simplifying the fraction, we get:
\[x = \frac{154}{20} = 7 \frac{14}{20}.\]
Finally, we can reduce the fraction by dividing both the numerator and the denominator by their greatest common divisor (which is $2$):
\[x = 7 \frac{14}{20} = 7 \frac{14 ÷ 2}{20 ÷ 2} = 7 \frac{7}{10}.\]
So, the length of the third side $BC$ is $\boxed{7 \frac{7}{10}}$ centimeters.
To find the length of the third side of the triangle, we can use the perimeter formula. The perimeter of a triangle is the sum of all its sides.
Given that the ant walks a total distance of 20 centimeters and two sides of the triangle are 5 1/10 centimeters and 7 1/5 centimeters, we can calculate the length of the third side.
Let's start by converting the mixed fractions into improper fractions:
5 1/10 centimeters can be written as (5 + 1/10) centimeters = (50/10 + 1/10) centimeters = 51/10 centimeters.
7 1/5 centimeters can be written as (7 + 1/5) centimeters = (35/5 + 1/5) centimeters = 36/5 centimeters.
Now, let's add all three sides together and set it equal to the total distance traveled by the ant:
51/10 + 36/5 + x = 20.
To solve for x, we need to find a common denominator. The least common multiple of 10 and 5 is 10.
First, multiply the numerator and denominator of 51/10 by 2 to get a common denominator of 10:
(51 * 2) / (10 * 2) = 102/20.
Similarly, multiply the numerator and denominator of 36/5 by 4 to get a common denominator of 20:
(36 * 4) / (5 * 4) = 144/20.
Now, we can rewrite the equation:
102/20 + 144/20 + x = 20.
Combining the fractions:
(102 + 144)/20 + x = 20.
246/20 + x = 20.
Next, let's simplify the left side by dividing both the numerator and denominator of 246/20 by their greatest common divisor, which is 2:
(246/2) / (20/2) + x = 20.
123/10 + x = 20.
To isolate x, subtract 123/10 from both sides of the equation:
x = 20 - 123/10.
To simplify the right side, let's find a common denominator for 20 and 123/10, which is 10.
To do this, multiply the numerator and denominator of 20 by 10:
20 * (10/10) = 200/10.
Now, we can rewrite the equation:
x = 200/10 - 123/10.
Subtract the numerators:
x = (200 - 123)/10.
x = 77/10.
Therefore, the length of the third side of the triangle is 7 7/10 centimeters or 77/10 centimeters.