Math  /  Numbers & Operations

Question14 books2 boxesand20 books3 boxes\frac{14 \text{ books}}{2 \text{ boxes}} \quad \text{and} \quad \frac{20 \text{ books}}{3 \text{ boxes}}
Tell whether the given rates form a proportion.

Studdy Solution

STEP 1

What is this asking? Are the two rates of books per box the same? Watch out! Don't just compare the number of books or boxes – it's the *rate* that matters!

STEP 2

1. Simplify the first rate
2. Simplify the second rate
3. Compare the simplified rates

STEP 3

Alright, let's break down the first rate: 14 books2 boxes\frac{14 \text{ books}}{2 \text{ boxes}}.
We want to find out how many books there are *per box*, so we'll **divide** the number of books by the number of boxes.
We're doing this to get a **unit rate**, which means the number of books for *one* box.

STEP 4

So, we have 14 books÷2 boxes=142booksbox14 \text{ books} \div 2 \text{ boxes} = \frac{14}{2} \frac{\text{books}}{\text{box}}.
Remember, dividing by 22 is the same as multiplying by 12\frac{1}{2}.
We're essentially asking, "What's half of 14 books?"

STEP 5

Calculating this gives us 142booksbox=7booksbox\frac{14}{2} \frac{\text{books}}{\text{box}} = 7 \frac{\text{books}}{\text{box}}.
Awesome! This means the first rate simplifies to **7 books per box**.

STEP 6

Now, let's tackle the second rate: 20 books3 boxes\frac{20 \text{ books}}{3 \text{ boxes}}.
Just like before, we want to find the number of books *per box*, so we'll **divide** the number of books by the number of boxes.

STEP 7

We have 20 books÷3 boxes=203booksbox20 \text{ books} \div 3 \text{ boxes} = \frac{20}{3} \frac{\text{books}}{\text{box}}.
This is like asking, "If we split 20 books evenly into 3 boxes, how many books would be in each box?"

STEP 8

This division gives us 203booksbox=623booksbox\frac{20}{3} \frac{\text{books}}{\text{box}} = 6 \frac{2}{3} \frac{\text{books}}{\text{box}}.
So, the second rate simplifies to **6 and two-thirds books per box**.

STEP 9

Okay, time to compare!
We found that the first rate is 7booksbox7 \frac{\text{books}}{\text{box}} and the second rate is 623booksbox6 \frac{2}{3} \frac{\text{books}}{\text{box}}.
Are these the same?
Nope! 77 is *not* equal to 6236 \frac{2}{3}.

STEP 10

Since the simplified rates are different, the rates do *not* form a proportion.

STEP 11

NO. The rates do not form a proportion.

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