Amelia is making brooches to sell at a craft fair. It takes her five minutes to make a star brooch and seven minutes to make a bow brooch. On Saturday, she spent four and three-fourths hours making brooches. She made three fewer star brooches than bow brooches. Write an equation that can be used to find 𝑦, the number of star brooches that Amelia made.
So in the question, we talk about two things: the amount of time it takes to make a brooch and how many brooches were made. Let’s first write an equation in terms of time. It takes her five minutes to make a star brooch. So we need to take the number of star brooches and multiply it by five.
Next, it says it takes her seven minutes to make a bow brooch. So we will be adding these together so that way we know how much time was spent. It says, on Saturday, she spent four and three-fourths hour making brooches.
However, there’s something wrong with this equation. The five and the seven are in terms of minutes, and four and three-fourths is in number of hours. So let’s go ahead and change the four and three-fourths to minutes. That way, the entire equation is the same. There are 60 minutes in one hour. So if we have four hours, we need to take 60 and multiply it by four, so that’s 240. But then we also have three-fourths of an hour, and 60 times three-fourths is 45. So four and three-fourths hours must be equal to 240 minutes plus 45 minutes, which is 285 minutes.
Now we can begin talking about the number of brooches made. She made three fewer star brooches than bow brooches. And we’re to use 𝑦 to represent the number of star brooches. So again, she made three fewer star brooches than bow brooches. So there were less star brooches.
So in order to find the number of star brooches, we would have to take the number of bow brooches and subtract three. So we know that star brooches can be represented by 𝑦. However, we need to know how to represent the number of bow brooches, and we need it to be in terms of 𝑦.
So we have the equation that star equals bow minus three. So replace star with 𝑦 and solve for bow. So we need to add three to both sides of the equation. So the number of bow brooches is equal to 𝑦 plus three. So to make one final equation, we can replace star with 𝑦 and bow with 𝑦 plus three, resulting in the equation five 𝑦 plus seven times 𝑦 plus three equals 285.