A second 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 third side of the triangle have length $x$ centimeters. Then the first ant walks along this third side, so it covers a distance of $x$ centimeters. The second ant walks along the other two sides of the triangle, so it covers a distance of $5 \frac{1}{10} + 7 \frac{1}{5}$ centimeters. Since this total distance is 20 centimeters, we have \[5 \frac{1}{10} + 7 \frac{1}{5} + x = 20.\]We can write the mixed numbers as fractions and then add: \begin{align*}
5 \frac{1}{10} + 7 \frac{1}{5} + x &= \frac{51}{10} + \frac{36}{5} + x \\
&= \frac{51}{10} + \frac{72}{10} + x \\
&= \frac{123}{10} + x.
\end{align*}Hence, we have \[\frac{123}{10} + x = 20,\]so $x = 20 - \frac{123}{10} = \boxed{\frac{77}{10}}$ centimeters.
A second 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? explain
Let the length of the third side of the triangle be $x$ centimeters.
The first ant walks along this third side, so it covers a distance of $x$ centimeters.
The second ant walks all the way around the triangle, so it covers the other two sides. The first side has a length of 5 1/10 centimeters and the second side has a length of 7 1/5 centimeters.
To find the total distance traveled by the second ant, we add the lengths of these two sides: $5 \frac{1}{10} + 7 \frac{1}{5}$. To add these mixed numbers, we find common denominators: $5 \frac{1}{10} = \frac{51}{10}$ and $7 \frac{1}{5} = \frac{36}{5}$.
Thus, the total distance traveled by the second ant is $\frac{51}{10} + \frac{36}{5}$. Adding these fractions, we get $\frac{123}{10}$.
Since the total distance traveled by the second ant is 20 centimeters, we have the equation $\frac{123}{10} + x = 20$.
To solve for $x$, we subtract $\frac{123}{10}$ from both sides of the equation: $x = 20 - \frac{123}{10} = \frac{200}{10} - \frac{123}{10} = \frac{77}{10}$.
Therefore, the length of the third side of the triangle is $\boxed{\frac{77}{10}}$ centimeters.
show me the answer to the question in numbers
The length of the third side of the triangle is $\frac{77}{10}$ centimeters, which is equal to $7.7$ centimeters.
wrong please explain
Apologies for the confusion. Let's start from the beginning.
Let the length of the third side of the triangle be $x$ centimeters.
The first ant walks along this third side, so it covers a distance of $x$ centimeters.
The second ant walks all the way around the triangle, so it covers the other two sides. The first side has a length of 5 1/10 centimeters and the second side has a length of 7 1/5 centimeters.
To find the total distance traveled by the second ant, we add the lengths of these two sides: $5 \frac{1}{10} + 7 \frac{1}{5}$. To add these mixed numbers, we find common denominators: $5 \frac{1}{10} = \frac{51}{10}$ and $7 \frac{1}{5} = \frac{36}{5}$.
Thus, the total distance traveled by the second ant is $\frac{51}{10} + \frac{36}{5}$. Adding these fractions, we get $\frac{51}{10} + \frac{72}{10} = \frac{123}{10}$.
Since the total distance traveled by the second ant is 20 centimeters, we have the equation $\frac{123}{10} + x = 20$.
To solve for $x$, we subtract $\frac{123}{10}$ from both sides of the equation: $x = 20 - \frac{123}{10} = \frac{200}{10} - \frac{123}{10} = \frac{77}{10}$.
Therefore, the length of the third side of the triangle is $\frac{77}{10}$ centimeters, which is equal to $\boxed{7.7}$ centimeters.
A second 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? explain and show answer
Let's start by finding the sum of the lengths of the two given sides.
The first side has a length of 5 1/10 centimeters, which is equal to $5 + \frac{1}{10}$ centimeters, or $\frac{51}{10}$ centimeters.
The second side has a length of 7 1/5 centimeters, which is equal to $7 + \frac{1}{5}$ centimeters, or $\frac{36}{5}$ centimeters.
So, the total distance covered by the second ant on the two given sides is $\frac{51}{10} + \frac{36}{5}$ centimeters.
To add these fractions, we need a common denominator. The least common multiple of 10 and 5 is 10, so we can rewrite the fractions as $\frac{51}{10} + \frac{72}{10}$.
Combining the numerators, we get $\frac{51+72}{10} = \frac{123}{10}$.
Now, let's set up an equation to find the length of the third side of the triangle. The total distance traveled by the second ant is given as 20 centimeters, so we can write $\frac{123}{10} + x = 20$.
To solve for x, we first subtract $\frac{123}{10}$ from both sides of the equation:
$x = 20 - \frac{123}{10} = \frac{200}{10} - \frac{123}{10} = \frac{77}{10}$.
Therefore, the length of the third side of the triangle is $\boxed{\frac{77}{10}}$ centimeters, or equivalently, $7.7$ centimeters.
To find the length of the third side of the triangle, we need to subtract the lengths of the other two sides from the total distance traveled by the ant.
Let's start by converting the mixed numbers to improper fractions for easier calculations:
5 1/10 = 51/10
7 1/5 = 36/5
Now, we can subtract the lengths of the known sides from the total distance traveled:
20 - (51/10 + 36/5)
To calculate this expression, we need to find a common denominator for the fractions. The least common multiple of 10 and 5 is 10. So, we can rewrite the expression as:
20 - (51/10 + (36/5) x (2/2))
20 - (51/10 + 72/10)
20 - (123/10)
Next, we need to subtract these fractions:
20 - (123/10) = (200/10) - (123/10) = 77/10
Finally, we can convert the fraction back to a mixed number:
77/10 = 7 7/10
Therefore, the length of the third side of the triangle is 7 7/10 centimeters.