Part a) Use 4.5 litres equals one gallon to convert 54 litres to gallons. Part b) 𝑔 gallons is equal to 𝑙 litres. Using the fact that 4.5 litres equals one gallon, write a formula for 𝑔 in terms of 𝑙.
We know that 4.5 litres equals one gallon. And so, we could double four and a half: 4.5 times two equals nine. We’ve gone from 4.5 litres to nine litres. And that would be equal to two gallons. This is saying that every nine litres equals two gallons. We notice that 54 is a multiple of nine. 54 divided by nine equals six. And that means there are six sets of nine in every 54.
If every nine litres is two gallons and we have six sets of nine litres, we can multiply six by two to find out that there are 12 gallons in every 54 litres. We can also consider this as a ratio of litres to gallons: four and a half to one, four and a half litres to one gallon. We multiply both sides by two and we get a scaled up ratio of nine to two. We’ll scale it up an additional time from nine to 54. And that means we multiply by six on both sides. And again, we see that 54 litres is equal to 12 gallons.
In part b, we need to write a formula for 𝑔 in terms of 𝑙. Here, we can again start with a ratio. We want the ratio of 𝑔 to 𝑙. That equals to one to four and a half. To now find 𝑔 in terms of 𝑙, we can rewrite our ratio as a fraction: 𝑔 over 𝑙. 𝑔 over 𝑙 equals one over four and a half.
We have a decimal in the denominator and we don’t like to do that. We can resolve this by multiplying the numerator and the denominator by two, which gives us 𝑔 over 𝑙 equals two over nine. To have 𝑔 in terms of 𝑙, we want 𝑔 by itself. And we can find that by multiplying both sides of the equation by 𝑙.
On the left side, we have 𝑔 divided by 𝑙 multiplied by 𝑙. And so, the 𝑙s cancel out, leaving us with just 𝑔. On the right side, we have two-ninths multiplied by 𝑙. Two-ninths multiplied by 𝑙 is two-ninths 𝑙. At this point, we’ve accomplished our goal: 𝑔 is written in terms of 𝑙; 𝑔 equals two-ninths 𝑙.
What this tells us is that if we know 𝑙, the number of litres, we can multiply that by two-ninths and it will return the number of gallons. We can perform a quick check by using what we got in part a. The number of gallons in 54 litres should be two-ninths times 54. We can simplify 54 over nine to be six over one. Six times two equals 12.
54 litres equals 12 gallons.