Math / Algebra

QuestionThere is an unknown number of students at the beginning of lunch. Then ... - Each student has 2 friends come sit with them, tripling the number of students. - When the bell rings, most of the students leave the lunchroom. One-fifth $(1 / 5)$ of the students stay to finish their lunches. - Ten (10) students join the students who are still in the lunchroom. - The total number of people in the lunchroom now is 22.
Create an equation that correctly represents the story above and can be used to find the number of students in the lunchroom at the beginning of lunch.
The equation is $\square$ The number of students who were in the lunchroom at the beginning of the story is $\square$

Studdy Solution

STEP 1

What is this asking? How many students were in the lunchroom at the beginning, given a series of changes in the number of students, ending with 22 students? Watch out! Don't mix up the order of operations!
It's easy to get confused about when the students tripled, when some left, and when the ten new students arrived.

STEP 2

1. Define the variable
2. Build the equation
3. Solve for $x$

STEP 3

Let's use $x$ to represent the initial number of students in the lunchroom.
This is what we're trying to find!

STEP 4

First, each student has two friends join them, which triples the number of students.
So, we now have $3 \cdot x$ or $3x$ students.

STEP 5

Next, only $\frac{1}{5}$ of these students stay.
So we multiply by $\frac{1}{5}$ : $\frac{1}{5} \cdot 3x = \frac{3x}{5}$ .

STEP 6

Then, ten more students join the remaining students.
We add 10: $\frac{3x}{5} + 10$ .

STEP 7

Finally, we know the total number of students is now 22.
So, our equation is: $\frac{3x}{5} + 10 = 22$ .

STEP 8

Subtract 10 from both sides of the equation to isolate the term with $x$ : $\frac{3x}{5} + 10 - 10 = 22 - 10$ $\frac{3x}{5} = 12$

STEP 9

To get rid of the fraction, we multiply both sides by 5: $5 \cdot \frac{3x}{5} = 12 \cdot 5$ $3x = 60$

STEP 10

Finally, divide both sides by 3 to find the value of $x$ : $\frac{3x}{3} = \frac{60}{3}$ $x = 20$

STEP 11

The equation that represents the story is $\frac{3x}{5} + 10 = 22$ .
There were 20 students in the lunchroom at the beginning of the story.

Was this helpful?

Studdy Solution

STEP 1

STEP 2

1. Define the variable
2. Build the equation
3. Solve for $x$

STEP 3

Let's use $x$ to represent the initial number of students in the lunchroom.
This is what we're trying to find!

STEP 4

First, each student has two friends join them, which triples the number of students.
So, we now have $3 \cdot x$ or $3x$ students.

STEP 5

Next, only $\frac{1}{5}$ of these students stay.
So we multiply by $\frac{1}{5}$ : $\frac{1}{5} \cdot 3x = \frac{3x}{5}$ .

STEP 6

Then, ten more students join the remaining students.
We add 10: $\frac{3x}{5} + 10$ .

STEP 7

Finally, we know the total number of students is now 22.
So, our equation is: $\frac{3x}{5} + 10 = 22$ .

STEP 8

Subtract 10 from both sides of the equation to isolate the term with $x$ : $\frac{3x}{5} + 10 - 10 = 22 - 10$ $\frac{3x}{5} = 12$

STEP 9

To get rid of the fraction, we multiply both sides by 5: $5 \cdot \frac{3x}{5} = 12 \cdot 5$ $3x = 60$

STEP 10

Finally, divide both sides by 3 to find the value of $x$ : $\frac{3x}{3} = \frac{60}{3}$ $x = 20$

STEP 11

The equation that represents the story is $\frac{3x}{5} + 10 = 22$ .
There were 20 students in the lunchroom at the beginning of the story.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Studdy Solution

STEP 1

STEP 2

1. Define the variable2. Build the equation3. Solve for xxx

STEP 3

Let's use xxx to represent the **initial number** of students in the lunchroom.This is what we're trying to find!

STEP 4

First, each student has two friends join them, which **triples** the number of students.So, we now have 3⋅x3 \cdot x3⋅x or 3x3x3x students.

STEP 5

Next, only 15\frac{1}{5}51​ of these students stay.So we multiply by 15\frac{1}{5}51​: 15⋅3x=3x5\frac{1}{5} \cdot 3x = \frac{3x}{5}51​⋅3x=53x​.

STEP 6

Then, **ten** more students join the remaining students.We **add** 10: 3x5+10\frac{3x}{5} + 1053x​+10.

STEP 7

Finally, we know the **total** number of students is now **22**.So, our equation is: 3x5+10=22\frac{3x}{5} + 10 = 2253x​+10=22.

STEP 8

**Subtract 10** from both sides of the equation to isolate the term with xxx: 3x5+10−10=22−10 \frac{3x}{5} + 10 - 10 = 22 - 10 53x​+10−10=22−10 3x5=12 \frac{3x}{5} = 12 53x​=12

STEP 9

To get rid of the fraction, we **multiply** both sides by **5**: 5⋅3x5=12⋅5 5 \cdot \frac{3x}{5} = 12 \cdot 5 5⋅53x​=12⋅5 3x=60 3x = 60 3x=60

STEP 10

Finally, **divide** both sides by **3** to find the value of xxx: 3x3=603 \frac{3x}{3} = \frac{60}{3} 33x​=360​ x=20 x = 20 x=20

STEP 11

The equation that represents the story is 3x5+10=22\frac{3x}{5} + 10 = 2253x​+10=22.There were **20** students in the lunchroom at the beginning of the story.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. Define the variable
2. Build the equation
3. Solve for $x$

Let's use $x$ to represent the initial number of students in the lunchroom.
This is what we're trying to find!

First, each student has two friends join them, which triples the number of students.
So, we now have $3 \cdot x$ or $3x$ students.

Next, only $\frac{1}{5}$ of these students stay.
So we multiply by $\frac{1}{5}$ : $\frac{1}{5} \cdot 3x = \frac{3x}{5}$ .

Then, ten more students join the remaining students.
We add 10: $\frac{3x}{5} + 10$ .

Finally, we know the total number of students is now 22.
So, our equation is: $\frac{3x}{5} + 10 = 22$ .

Subtract 10 from both sides of the equation to isolate the term with $x$ : $\frac{3x}{5} + 10 - 10 = 22 - 10$ $\frac{3x}{5} = 12$

To get rid of the fraction, we multiply both sides by 5: $5 \cdot \frac{3x}{5} = 12 \cdot 5$ $3x = 60$

Finally, divide both sides by 3 to find the value of $x$ : $\frac{3x}{3} = \frac{60}{3}$ $x = 20$

The equation that represents the story is $\frac{3x}{5} + 10 = 22$ .
There were 20 students in the lunchroom at the beginning of the story.