Math Snap

STEP 1

What is this asking?
Which of these lists of fractions is in the correct order, from smallest to largest or largest to smallest?
Watch out!
Don't just guess based on how the fractions look.
We need to actually compare them!

STEP 2

1. Convert to a Common Denominator
2. Compare the Fractions
3. Determine the Correct Order

STEP 3

Let's find the least common multiple (LCM) of the denominators: 8, 3, and 4.
The LCM is the smallest number that all three denominators divide into evenly.
We can list multiples of each number until we find a common one.
Multiples of 8 are 8, 16, 24, 32... Multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, 27... Multiples of 4 are 4, 8, 12, 16, 20, 24, 28... So, our LCM is 24!

STEP 4

Now, we'll convert each fraction so it has a denominator of 24.
For $\frac{13}{8}$, we multiply the top and bottom by 3 (because $8 \cdot 3 = 24$): $\frac{13}{8} \cdot \frac{3}{3} = \frac{39}{24}$.
For $\frac{5}{3}$, we multiply the top and bottom by 8 (because $3 \cdot 8 = 24$): $\frac{5}{3} \cdot \frac{8}{8} = \frac{40}{24}$.
For $\frac{7}{4}$, we multiply the top and bottom by 6 (because $4 \cdot 6 = 24$): $\frac{7}{4} \cdot \frac{6}{6} = \frac{42}{24}$.

STEP 5

Now that all the fractions have the same denominator (24), we can compare them by looking at their numerators.
We have $\frac{39}{24}$, $\frac{40}{24}$, and $\frac{42}{24}$.

STEP 6

The numerators are 39, 40, and 42.
Since $39 < 40 < 42$, we know that $\frac{39}{24} < \frac{40}{24} < \frac{42}{24}$.

STEP 7

Substituting back the original fractions, we get $\frac{13}{8} < \frac{5}{3} < \frac{7}{4}$.

STEP 8

This matches option A!

The correct expression is A: $\frac{13}{8} < \frac{5}{3} < \frac{7}{4}$.