Math / Algebra

QuestionThe ratio of girls to boys in your class is 5 to 7 . Two girls joined your class. Now the ratio of girls to boys is 6 to 7 . How many students are now in your class? Explain how you got your answer.

Studdy Solution

STEP 1

What is this asking? If the ratio of girls to boys is 5 to 7, and then it changes to 6 to 7 after 2 girls join, how many total students are there now? Watch out! Don't forget that the number of boys doesn't change!

STEP 2

1. Set up the initial ratio.
2. Set up the new ratio.
3. Find the number of boys.
4. Calculate the total number of students.

STEP 3

Initially, the ratio of girls to boys is 5 to 7.
This means for every 5 girls, there are 7 boys.
We can write this as a fraction: $\frac{\text{Number of girls}}{\text{Number of boys}} = \frac{5}{7}$ .
Since we don't know the actual number of girls and boys, we'll use a variable to represent a scaling factor.
Let's use $x$ !

STEP 4

So, the number of girls is $5x$ and the number of boys is $7x$ .
This keeps the ratio at 5 to 7, no matter what $x$ is!
For example, if $x = 1$ , there are 5 girls and 7 boys.
If $x = 2$ , there are 10 girls and 14 boys, and the ratio is still $\frac{10}{14} = \frac{5}{7}$ !

STEP 5

Two girls join the class.
Awesome! Now the number of girls is $5x + 2$ .
The number of boys stays the same at $7x$ .
The new ratio is 6 to 7, so we can write this as $\frac{5x + 2}{7x} = \frac{6}{7}$ .

STEP 6

Now we have an equation we can solve! Multiply both sides of $\frac{5x + 2}{7x} = \frac{6}{7}$ by $7x$ to get $5x + 2 = 6x$ .
We're multiplying by $7x$ because it's the denominator on both sides, and we want to get rid of those fractions.
Remember, we're dividing by $7x$ on the left and multiplying by $7x$ , which is the same as multiplying by one.
And on the right, we're also multiplying and dividing by $7x$ , which is also the same as multiplying by one.

STEP 7

Subtract $5x$ from both sides of $5x + 2 = 6x$ to get $2 = x$ .
We're subtracting $5x$ from both sides to isolate $x$ and find its value.
Great, we found our scaling factor!

STEP 8

Since the number of boys is $7x$ , we substitute $x = 2$ to find that there are $7 \cdot 2 = 14$ boys.

STEP 9

The new number of girls is $5x + 2$ .
With $x = 2$ , we have $5 \cdot 2 + 2 = 10 + 2 = 12$ girls.

STEP 10

Now, add the number of girls and boys to find the total number of students: $12 + 14 = 26$ students.

STEP 11

There are now 26 students in the class.

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QuestionThe ratio of girls to boys in your class is 5 to 7 . Two girls joined your class. Now the ratio of girls to boys is 6 to 7 . How many students are now in your class? Explain how you got your answer.

Studdy Solution

STEP 1

What is this asking? If the ratio of girls to boys is 5 to 7, and then it changes to 6 to 7 after 2 girls join, how many total students are there now? Watch out! Don't forget that the number of boys doesn't change!

STEP 2

1. Set up the initial ratio.
2. Set up the new ratio.
3. Find the number of boys.
4. Calculate the total number of students.

STEP 3

STEP 4

So, the number of girls is $5x$ and the number of boys is $7x$ .
This keeps the ratio at 5 to 7, no matter what $x$ is!
For example, if $x = 1$ , there are 5 girls and 7 boys.
If $x = 2$ , there are 10 girls and 14 boys, and the ratio is still $\frac{10}{14} = \frac{5}{7}$ !

STEP 5

Two girls join the class.
Awesome! Now the number of girls is $5x + 2$ .
The number of boys stays the same at $7x$ .
The new ratio is 6 to 7, so we can write this as $\frac{5x + 2}{7x} = \frac{6}{7}$ .

STEP 6

STEP 7

Subtract $5x$ from both sides of $5x + 2 = 6x$ to get $2 = x$ .
We're subtracting $5x$ from both sides to isolate $x$ and find its value.
Great, we found our scaling factor!

STEP 8

Since the number of boys is $7x$ , we substitute $x = 2$ to find that there are $7 \cdot 2 = 14$ boys.

STEP 9

The new number of girls is $5x + 2$ .
With $x = 2$ , we have $5 \cdot 2 + 2 = 10 + 2 = 12$ girls.

STEP 10

Now, add the number of girls and boys to find the total number of students: $12 + 14 = 26$ students.

STEP 11

There are now 26 students in the class.

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QuestionThe ratio of girls to boys in your class is 5 to 7 . Two girls joined your class. Now the ratio of girls to boys is 6 to 7 . How many students are now in your class? Explain how you got your answer.

Studdy Solution

STEP 1

What is this asking? If the ratio of girls to boys is 5 to 7, and then it changes to 6 to 7 after 2 girls join, how many total students are there now? Watch out! Don't forget that the number of boys *doesn't change*!

STEP 2

1. Set up the initial ratio.2. Set up the new ratio.3. Find the number of boys.4. Calculate the total number of students.

STEP 3

STEP 4

So, the number of girls is 5x5x5x and the number of boys is 7x7x7x.This keeps the ratio at 5 to 7, no matter what xxx is!For example, if x=1x = 1x=1, there are 5 girls and 7 boys.If x=2x = 2x=2, there are 10 girls and 14 boys, and the ratio is still 1014=57\frac{10}{14} = \frac{5}{7}1410​=75​!

STEP 5

Two girls join the class.Awesome! Now the number of girls is 5x+25x + 25x+2.The number of boys *stays the same* at 7x7x7x.The *new* ratio is 6 to 7, so we can write this as 5x+27x=67\frac{5x + 2}{7x} = \frac{6}{7}7x5x+2​=76​.

STEP 6

STEP 7

**Subtract** 5x5x5x from both sides of 5x+2=6x5x + 2 = 6x5x+2=6x to get 2=x2 = x2=x.We're subtracting 5x5x5x from both sides to isolate xxx and find its value.Great, we found our scaling factor!

STEP 8

Since the number of boys is 7x7x7x, we **substitute** x=2x = 2x=2 to find that there are 7⋅2=147 \cdot 2 = 147⋅2=14 boys.

STEP 9

The new number of girls is 5x+25x + 25x+2.With x=2x = 2x=2, we have 5⋅2+2=10+2=125 \cdot 2 + 2 = 10 + 2 = 125⋅2+2=10+2=12 girls.

STEP 10

Now, **add** the number of girls and boys to find the total number of students: 12+14=2612 + 14 = 2612+14=26 students.

STEP 11

There are now **26** students in the class.

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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.

What is this asking? If the ratio of girls to boys is 5 to 7, and then it changes to 6 to 7 after 2 girls join, how many total students are there now? Watch out! Don't forget that the number of boys doesn't change!

1. Set up the initial ratio.
2. Set up the new ratio.
3. Find the number of boys.
4. Calculate the total number of students.

So, the number of girls is $5x$ and the number of boys is $7x$ .
This keeps the ratio at 5 to 7, no matter what $x$ is!
For example, if $x = 1$ , there are 5 girls and 7 boys.
If $x = 2$ , there are 10 girls and 14 boys, and the ratio is still $\frac{10}{14} = \frac{5}{7}$ !

Two girls join the class.
Awesome! Now the number of girls is $5x + 2$ .
The number of boys stays the same at $7x$ .
The new ratio is 6 to 7, so we can write this as $\frac{5x + 2}{7x} = \frac{6}{7}$ .

Subtract $5x$ from both sides of $5x + 2 = 6x$ to get $2 = x$ .
We're subtracting $5x$ from both sides to isolate $x$ and find its value.
Great, we found our scaling factor!

Since the number of boys is $7x$ , we substitute $x = 2$ to find that there are $7 \cdot 2 = 14$ boys.

The new number of girls is $5x + 2$ .
With $x = 2$ , we have $5 \cdot 2 + 2 = 10 + 2 = 12$ girls.

Now, add the number of girls and boys to find the total number of students: $12 + 14 = 26$ students.

There are now 26 students in the class.