Problem 78 - Algebra 1: Concepts and Skills

Question

Complete the statement using \(<,>,\) or \(=.\) (Skills Review pp. \(763,770,771\) ) $$\frac{8}{7} ? 1 \frac{1}{7}$$

Step-by-Step Solution

Answer

As such the statement is \( \frac{8}{7} = 1 \frac{1}{7} \).

Step 1: Change the mixed fraction to an improper fraction

An improper fraction is a fraction in which the numerator (top number) is greater than the denominator (bottom number). To convert mixed number to an improper fraction, use the general rule: multiply the denominator of the fractional part by the whole number, then add the numerator to get your new numerator. The denominator stays the same. Thus, the number \( 1 \frac{1}{7} \) converted to an improper fraction is \( \frac{8}{7} \). So our comparison now is \( \frac{8}{7} ? \frac{8}{7} \).

Step 2: Compare the fractions

Since the fractions are identical, i.e. \( \frac{8}{7} \) is equal to \( \frac{8}{7} \), then the correct symbol to use in the question is \( = \).

Key Concepts

Improper Fractions

Improper fractions might sound like they're breaking math rules, but they're totally legit! They're just fractions where the top number (the numerator) is bigger than the bottom number (the denominator). Think of it like having more pizza than you were planning for a party - it's more than a whole pizza!

For example, if you see something like \(\frac{8}{7}\), that's an improper fraction. To make sense of it, imagine you have 8 slices of pizza, but a whole pizza only has 7 slices. So, you've got more than one full pizza here. Improper fractions are really useful in math, especially when you're working with mixed numbers or comparing fractions.

Mixed Numbers

Mixed numbers are like a party with whole pizzas and extra slices on the side. They've got a whole number part and a fraction part, all hanging out together. For instance, in \(1\frac{1}{7}\), you've got one whole pizza, plus one of those 7 slices from another pizza.

To turn a mixed number into an improper fraction (so it's easier to use in math), you do a little multiplication and addition. You multiply the whole number by the denominator (the bottom part of the fraction) and then add the numerator (the top part of the fraction). So, with \(1\frac{1}{7}\), you do \(1\times 7 + 1\) which gives you the improper fraction \(\frac{8}{7}\). This way, you can compare it to other fractions without breaking a sweat.

Fraction Comparison

When you compare fractions, you're basically looking to see which one is bigger, or if they're twinsies (equal). It's crucial when you're trying to find out who has more pizza, for example. If fractions have the same bottom number (denominator), like \(\frac{8}{7}\) and \(\frac{8}{7}\), you're in luck because you can just look at the top numbers (numerators). Since the numerators are the same here, it's like having two identical pizzas. They're equal, which in math language, we write as \(\frac{8}{7} = \frac{8}{7}\).

But if they're different, you've got a bit more work to do - you might have to turn mixed numbers into improper fractions like we talked about before, or find a common denominator. The goal is to get fractions that you can directly compare, making your math life a lot easier!