Solving equations and inequalities through substitution example 3

Joey is training for a hot dog eating contest. The person who eats the most hot dogs in 10 minutes is the winner. If r is the number of hot dogs that Joey can eat in a minute and n is the total number of hot dogs he eats in the contest, we can write the following equation.

So, this just makes sense: he eats a total of n hot dogs, and the contest is 10 minutes, so you divide that by 10. He's going to eat n hot dogs in 10 minutes, and so yes, this will give you the number of how many hot dogs on average he ate per minute.

If Joey can eat six and four-fifths hot dogs per minute, so they're saying r is six and four-fifths. How many hot dogs does he eat during the ten-minute contest? So, they're saying, hey, what is n going to be right over here?

N divided by 10 is six and four-fifths. R in this situation is six and four-fifths; that’s how many hot dogs he can eat per minute. Now, there are a couple of ways that you could tackle this.

You could say, hey, some number divided by 10 is six and four-fifths, and so you say, well, that number must be six and four-fifths times ten. Six and four-fifths times ten would be 60 plus 8 or 68. So that's one way you could do it. You're essentially multiplying both sides of this equation by 10.

The other way you could do it is you could just try out numbers, and that usually takes a lot longer. But at least it'll give you the sense of how the equation relates, and especially because they have choices here, it's not a bad idea.

If obviously the ideal one is to just think logically through it. Hey, if something divided by ten is six and four-fifths, then ten times six and four-fifths is going to be that something, and that's how you could get 68.

Or if you want, you could try out the different numbers. Look, 65 divided by 10 is only going to give you 6.5; that's not the r we're looking for. 56 divided by 10 is going to give you 5.6; not the r we're looking for.

68 divided by 10 is going to give you six point eight six or six and eight tenths, and six point and six and eight tenths is the same thing as six and four-fifths. So 68 works.

45 hot dogs, you divide that by 10, you're going to get 4.5 hot dogs per minute. So all of these are completely legitimate ways of doing it. The important thing is that you understand this relationship and you understand how these variables work with respect to each other.