Math / Algebra

Question2. A car's gas tank can hold 14 gallons of gas. A mechanic determines that the tank is $13 \%$ full. a. Write and solve an equation of the form $y=k x$ to determine the number of gallons of gas in the tank. b. Write and solve a proportion to determine the number of gallons of gas in the tank.

Studdy Solution

STEP 1

What is this asking? We need to figure out how much gas is in a car's tank if it's only 13% full and can hold 14 gallons.
We'll do this two ways: with an equation and with a proportion. Watch out! Percentages can be tricky!
Remember that 13% means 13 out of 100.
Don't mix up the percentage with the actual amount of gas.

STEP 2

1. Solve with an equation
2. Solve with a proportion

STEP 3

Let $y$ be the number of gallons of gas in the tank, and let $x$ be the total capacity of the tank, which is 14 gallons. $k$ represents the percentage full, which is 13% or $\frac{13}{100}$ as a decimal, this is 0.13.

STEP 4

Our equation is $y = k \cdot x$ .
This makes sense because the amount of gas in the tank ( $y$ ) is a fraction ( $k$ ) of the total capacity ( $x$ ).

STEP 5

We know $k = \mathbf{0.13}$ and $x = \mathbf{14}$ , so we substitute those values into our equation: $y = \mathbf{0.13} \cdot \mathbf{14}$ .

STEP 6

Multiplying $\mathbf{0.13}$ by $\mathbf{14}$ gives us $y = \mathbf{1.82}$ .
So there are 1.82 gallons of gas in the tank!

STEP 7

A proportion compares two ratios.
We can set up a proportion like this: $\frac{\text{gallons of gas}}{\text{total capacity}} = \frac{\text{percentage}}{\mathbf{100}}$ .

STEP 8

We know the total capacity is 14 gallons, and the percentage full is 13%.
Let $y$ represent the gallons of gas.
Our proportion becomes $\frac{y}{\mathbf{14}} = \frac{\mathbf{13}}{\mathbf{100}}$ .

STEP 9

To solve for $y$ , we want to isolate it.
We can do this by multiplying both sides of the equation by 14: $y = \frac{\mathbf{13}}{\mathbf{100}} \cdot \mathbf{14}$ .

STEP 10

Multiplying $\frac{\mathbf{13}}{\mathbf{100}}$ by $\mathbf{14}$ gives us $y = \mathbf{1.82}$ .
This confirms our earlier result!
There are 1.82 gallons of gas in the tank.

STEP 11

The car has 1.82 gallons of gas in its tank.
We found this using both an equation of the form $y=k \cdot x$ and by setting up a proportion.

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Studdy Solution

STEP 1

STEP 2

1. Solve with an equation
2. Solve with a proportion

STEP 3

Let $y$ be the number of gallons of gas in the tank, and let $x$ be the total capacity of the tank, which is 14 gallons. $k$ represents the percentage full, which is 13% or $\frac{13}{100}$ as a decimal, this is 0.13.

STEP 4

Our equation is $y = k \cdot x$ .
This makes sense because the amount of gas in the tank ( $y$ ) is a fraction ( $k$ ) of the total capacity ( $x$ ).

STEP 5

We know $k = \mathbf{0.13}$ and $x = \mathbf{14}$ , so we substitute those values into our equation: $y = \mathbf{0.13} \cdot \mathbf{14}$ .

STEP 6

Multiplying $\mathbf{0.13}$ by $\mathbf{14}$ gives us $y = \mathbf{1.82}$ .
So there are 1.82 gallons of gas in the tank!

STEP 7

A proportion compares two ratios.
We can set up a proportion like this: $\frac{\text{gallons of gas}}{\text{total capacity}} = \frac{\text{percentage}}{\mathbf{100}}$ .

STEP 8

We know the total capacity is 14 gallons, and the percentage full is 13%.
Let $y$ represent the gallons of gas.
Our proportion becomes $\frac{y}{\mathbf{14}} = \frac{\mathbf{13}}{\mathbf{100}}$ .

STEP 9

To solve for $y$ , we want to isolate it.
We can do this by multiplying both sides of the equation by 14: $y = \frac{\mathbf{13}}{\mathbf{100}} \cdot \mathbf{14}$ .

STEP 10

Multiplying $\frac{\mathbf{13}}{\mathbf{100}}$ by $\mathbf{14}$ gives us $y = \mathbf{1.82}$ .
This confirms our earlier result!
There are 1.82 gallons of gas in the tank.

STEP 11

The car has 1.82 gallons of gas in its tank.
We found this using both an equation of the form $y=k \cdot x$ and by setting up a proportion.

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Studdy Solution

STEP 1

STEP 2

1. Solve with an equation2. Solve with a proportion

STEP 3

Let yyy be the number of gallons of gas in the tank, and let xxx be the total capacity of the tank, which is **14 gallons**. kkk represents the percentage full, which is **13%** or 13100\frac{13}{100}10013​ as a decimal, this is **0.13**.

STEP 4

Our equation is y=k⋅xy = k \cdot xy=k⋅x.This makes sense because the amount of gas in the tank (yyy) is a fraction (kkk) of the total capacity (xxx).

STEP 5

We know k=0.13k = \mathbf{0.13}k=0.13 and x=14x = \mathbf{14}x=14, so we substitute those values into our equation: y=0.13⋅14y = \mathbf{0.13} \cdot \mathbf{14}y=0.13⋅14.

STEP 6

Multiplying 0.13\mathbf{0.13}0.13 by 14\mathbf{14}14 gives us y=1.82y = \mathbf{1.82}y=1.82.So there are **1.82 gallons** of gas in the tank!

STEP 7

A proportion compares two ratios.We can set up a proportion like this: gallons of gastotal capacity=percentage100\frac{\text{gallons of gas}}{\text{total capacity}} = \frac{\text{percentage}}{\mathbf{100}}total capacitygallons of gas​=100percentage​.

STEP 8

We know the total capacity is **14 gallons**, and the percentage full is **13%**.Let yyy represent the gallons of gas.Our proportion becomes y14=13100\frac{y}{\mathbf{14}} = \frac{\mathbf{13}}{\mathbf{100}}14y​=10013​.

STEP 9

To solve for yyy, we want to isolate it.We can do this by multiplying both sides of the equation by **14**: y=13100⋅14y = \frac{\mathbf{13}}{\mathbf{100}} \cdot \mathbf{14}y=10013​⋅14.

STEP 10

Multiplying 13100\frac{\mathbf{13}}{\mathbf{100}}10013​ by 14\mathbf{14}14 gives us y=1.82y = \mathbf{1.82}y=1.82.This confirms our earlier result!There are **1.82 gallons** of gas in the tank.

STEP 11

The car has **1.82 gallons** of gas in its tank.We found this using both an equation of the form y=k⋅xy=k \cdot xy=k⋅x and by setting up a proportion.

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1. Solve with an equation
2. Solve with a proportion

Let $y$ be the number of gallons of gas in the tank, and let $x$ be the total capacity of the tank, which is 14 gallons. $k$ represents the percentage full, which is 13% or $\frac{13}{100}$ as a decimal, this is 0.13.

Our equation is $y = k \cdot x$ .
This makes sense because the amount of gas in the tank ( $y$ ) is a fraction ( $k$ ) of the total capacity ( $x$ ).

We know $k = \mathbf{0.13}$ and $x = \mathbf{14}$ , so we substitute those values into our equation: $y = \mathbf{0.13} \cdot \mathbf{14}$ .

Multiplying $\mathbf{0.13}$ by $\mathbf{14}$ gives us $y = \mathbf{1.82}$ .
So there are 1.82 gallons of gas in the tank!

A proportion compares two ratios.
We can set up a proportion like this: $\frac{\text{gallons of gas}}{\text{total capacity}} = \frac{\text{percentage}}{\mathbf{100}}$ .

We know the total capacity is 14 gallons, and the percentage full is 13%.
Let $y$ represent the gallons of gas.
Our proportion becomes $\frac{y}{\mathbf{14}} = \frac{\mathbf{13}}{\mathbf{100}}$ .

To solve for $y$ , we want to isolate it.
We can do this by multiplying both sides of the equation by 14: $y = \frac{\mathbf{13}}{\mathbf{100}} \cdot \mathbf{14}$ .

Multiplying $\frac{\mathbf{13}}{\mathbf{100}}$ by $\mathbf{14}$ gives us $y = \mathbf{1.82}$ .
This confirms our earlier result!
There are 1.82 gallons of gas in the tank.

The car has 1.82 gallons of gas in its tank.
We found this using both an equation of the form $y=k \cdot x$ and by setting up a proportion.