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Math  /  Algebra

Question6. Una maçuina envasa 1200 latas de refresco en 8 horas. Cuántas horas se demora en envasar: a) 50 latas b) 100 latas c) 200 latas d) 100∗[100^{*}[ ] latas

Studdy Solution

STEP 1

What is this asking? If a machine packages 1200 cans in 8 hours, how long does it take to package 50, 100, 200, and an unknown number of cans? Watch out! Make sure to keep track of your units (cans and hours)!

STEP 2

1. Calculate the rate.
2. Calculate the time for 50 cans.
3. Calculate the time for 100 cans.
4. Calculate the time for 200 cans.
5. Calculate the time for an unknown number of cans.

STEP 3

Alright, let's **start** by figuring out how many cans this machine packages per hour.
We know it packages 1200 cans\text{1200 cans} in 8 hours\text{8 hours}.

STEP 4

To find the rate, we'll **divide** the number of cans by the number of hours.
This tells us how fast the machine is working! 1200 cans8 hours=150 cans1 hour \frac{\text{1200 cans}}{\text{8 hours}} = \frac{\text{150 cans}}{\text{1 hour}} So, the machine packages 150 cans per hour\textbf{150 cans per hour}!

STEP 5

Now, we want to know how long it takes to package 50 cans\text{50 cans}.
We know the machine packages 150 cans\text{150 cans} in 1 hour\text{1 hour}.

STEP 6

We can **set up a proportion**: 150 cans1 hour=50 cansx hours \frac{\text{150 cans}}{\text{1 hour}} = \frac{\text{50 cans}}{\text{x hours}}

STEP 7

**Cross-multiply**: 150 cans⋅x hours=50 cans⋅1 hour \text{150 cans} \cdot \text{x hours} = \text{50 cans} \cdot \text{1 hour}

STEP 8

**Solve for** xx: x=50 cans⋅1 hour150 cans=13 hour x = \frac{\text{50 cans} \cdot \text{1 hour}}{\text{150 cans}} = \frac{1}{3} \text{ hour} It takes 1/3 of an hour\textbf{1/3 of an hour} to package 50 cans!

STEP 9

Let's do the same thing for 100 cans\text{100 cans}.

STEP 10

**Proportion**: 150 cans1 hour=100 cansx hours \frac{\text{150 cans}}{\text{1 hour}} = \frac{\text{100 cans}}{\text{x hours}}

STEP 11

**Cross-multiply and solve**: x=100 cans⋅1 hour150 cans=23 hour x = \frac{\text{100 cans} \cdot \text{1 hour}}{\text{150 cans}} = \frac{2}{3} \text{ hour} It takes 2/3 of an hour\textbf{2/3 of an hour} to package 100 cans!

STEP 12

Now for 200 cans\text{200 cans}!

STEP 13

**Proportion**: 150 cans1 hour=200 cansx hours \frac{\text{150 cans}}{\text{1 hour}} = \frac{\text{200 cans}}{\text{x hours}}

STEP 14

**Cross-multiply and solve**: x=200 cans⋅1 hour150 cans=43 hour x = \frac{\text{200 cans} \cdot \text{1 hour}}{\text{150 cans}} = \frac{4}{3} \text{ hour} It takes 4/3 of an hour\textbf{4/3 of an hour} (or 1 hour and 20 minutes) to package 200 cans!

STEP 15

Looks like there's a typo in the problem.
Let's assume it meant to ask how long it takes to package 100\text{100} cans multiplied by some unknown number, which we'll call nn.

STEP 16

So, we want to find the time to package 100⋅n100 \cdot n cans.

STEP 17

**Proportion**: 150 cans1 hour=100⋅n cansx hours \frac{\text{150 cans}}{\text{1 hour}} = \frac{\text{100} \cdot n \text{ cans}}{\text{x hours}}

STEP 18

**Cross-multiply and solve**: x=100⋅n cans⋅1 hour150 cans=2n3 hour x = \frac{\text{100} \cdot n \text{ cans} \cdot \text{1 hour}}{\text{150 cans}} = \frac{2n}{3} \text{ hour} It takes (2n/3) hours\textbf{(2n/3) hours} to package 100⋅n100 \cdot n cans!

STEP 19

a) 1/31/3 hour b) 2/32/3 hour c) 4/34/3 hours d) (2n/3)(2n/3) hours (assuming the question meant 100⋅n100 \cdot n cans)

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