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Chapter 1: Problem 402

Simplify. $$ \frac{7^{2}+1}{60} $$

Short Answer

Expert verified
\(\frac{5}{6}\).

Step by step solution

01

Calculate the numerator

First, calculate the value of the expression in the numerator, which is \(7^{2} + 1\).Compute \(7^{2}\):\(7^{2} = 49\)Then add 1:\(49 + 1 = 50\).So, the numerator is 50.
02

Write the simplified fraction

Now, substitute the value of the numerator back into the fraction:\(\frac{7^{2} + 1}{60} = \frac{50}{60}\).
03

Simplify the fraction

Simplify the fraction \(\frac{50}{60}\) by finding the greatest common divisor (GCD) of 50 and 60, which is 10.Divide the numerator and denominator by the GCD (10):\[\frac{50 \div 10}{60 \div 10} = \frac{5}{6}\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

numerator calculation
When working with fractions, understanding how to calculate the numerator is fundamental. The numerator is the top part of a fraction and it represents how many parts of the whole are being considered. In our example, we start with the expression \(7^{2} + 1\). Calculating the numerator involves evaluating this expression step-by-step: \(7^{2}\) means multiplying 7 by itself which is \(49\). Then, follow by adding 1 to 49, giving us \(50\). Therefore, the numerator of our fraction is \(50\). Remember, breaking down the calculation into smaller steps can make it easier to follow and understand.
greatest common divisor
The greatest common divisor (GCD) is essential for simplifying fractions. It is the largest number that can evenly divide both the numerator and the denominator. To find the GCD of two numbers, you can list their factors and identify the highest number common to both. For the given fraction \( \frac{50}{60} \), the factors of 50 are \[ \text{1, 2, 5, 10, 25, 50} \] and the factors of 60 are \[ \text{1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60} \]. The highest common factor is 10, hence, the GCD of 50 and 60 is 10.
simplifying fractions
To simplify a fraction, divide both the numerator and the denominator by their greatest common divisor (GCD). In our example with the fraction \( \frac{50}{60} \), we identified the GCD as 10. Now, divide both the numerator (50) and the denominator (60) by 10. When you do this division: \[ \frac{50 \div 10}{60 \div 10} = \frac{5}{6} \]. The fraction is now simplified to \( \frac{5}{6} \). Simplifying fractions makes them easier to work with in further calculations and comparisons. Always check if both the numerator and the denominator can be divided by any common factor to reduce the fraction to its simplest form.

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