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A car can travel 476 miles on 14 gallons of gas. Write an equation relating the distance $d$ to the number of gallons $g$. How many gallons of gas does this car need to travel 578 miles?

Last updated date: 12th Sep 2024
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Hint: In order to express a mathematical problem explained in words we have to take all the relevant and important information mentioned in the word problem and represent it in a much easier way like an equation.
After representing it in the symbolic form we have to try to eliminate various variables represented using equations by rearranging terms, using basic arithmetic operations and substitutions. Eventually we would be able to solve the question. So by using the above definition and techniques we can solve the given question.

Complete step by step solution:
Given statement
${\text{On travelling }}476{\text{ miles the car consumes }}14{\text{ gallons of gas}}.................................\left( i \right)$
Now we need to write the equation relating the distance $d$ to the number of gallons $g$.
Also the given statement can be expressed in terms of $d\;{\text{and}}\;g$ as below:
$476 \times d = 14 \times g.............\left( {ii} \right)$
Therefore the equation relating the distance $d$ to the number of gallons $g$ can be written as:
$476 \times d = 14 \times g$
Now we need to find the gallons of gas this car needs to travel 578 miles:
Such that we now that:
$476 \times d = 14 \times g$
On dividing (ii) with $476$ on both LHS and RHS we can find how much gallons of gas is needed for
$1\;{\text{mile}}$ such that we get:
$476 \times d = 14 \times g \\ \dfrac{{476 \times d}}{{476}} = \dfrac{{14 \times g}}{{476}} \\ d = \dfrac{1}{{34}}g..................\left( {iii} \right) \\$
From (iii) we can say that $1\;{\text{mile}}$ requires $\dfrac{1}{{34}}$ gallons of gas.
Here we need to find the gallons of gas this car needs to travel 578 miles, such that we need to just multiply 578 to both LHS and RHS in the equation (iii).
Such that we get:
$d = \dfrac{1}{{34}}g \\ 578d = \dfrac{{578}}{{34}}g \\ 578d = 17g...................\left( {iv} \right) \\$
So from (iv) we can write that this car needs:$17{\text{ gallons of gas to travel }}578{\text{ miles}}$

Note: On solving similar word problems, it’s advisable to adopt the above prescribed method since it’s easy to understand and execute. Also while solving word problems it’s important to read the question properly. After reading the question properly one has to separate the relevant and irrelevant information carefully.