Part a) A group of 𝑆 students visit a corner shop. Each student buys 𝐺 gummy bears from the shop. Write an expression for the total number of gummy bears the students have bought.
So we’re told in the question that the number of students is 𝑆 and each student buys the same amount of gummy bears. They each buy 𝐺 gummy bears. This means that the number of gummy bears bought is equal to 𝑆 lots of 𝐺.
Now to find, for example, five lots of two, we would multiply five and two together. So to find 𝑆 lots of 𝐺, we would multiply 𝑆 by 𝐺. However, in an algebraic expression, we don’t need to include the multiplication sign, so 𝑆 multiplied by 𝐺 can just be written as 𝑆𝐺. This is our expression for the total number of gummy bears bought by the group of students.
Now it’s just worth noting that another acceptable answer would be 𝐺𝑆, as it doesn’t matter which way around we write the letters in an algebraic product. In fact, it’s actually more usual to write the letters in alphabetical order.
Part b) Solve the equation six 𝑥 plus four is equal to 20.
To solve this equation, we need to find the value of 𝑥 that makes this equation true. We look at two methods for solving this equation. The first is a formal algebraic approach. Whenever we solve an equation, we need to make sure that we perform the same operations or steps to both sides of the equation.
Now 𝑥 is on the left of the equation, so let’s see what we need to do in order to leave 𝑥 on its own. There’s a plus four here, so our first step is going to be to subtract four in order to cancel this out. But whatever we do to one side of the equation we must also do to the other side. So we’re subtracting four from both sides.
On the left of the equation, we’re just left with six 𝑥, as the plus four and the minus four cancel each other out. And on the right, we have 20 minus four, which is equal to 16. So our equation becomes six 𝑥 equals 16.
Now we want to work out what 𝑥 is equal to, but at the moment we have six 𝑥, which remember means six multiplied by 𝑥. So to leave us with 𝑥 on its own, we need to divide both sides of the equation by six. Dividing six 𝑥 by six just gives 𝑥, and dividing 16 by six, well we can write this as the fraction 16 over six.
Now we’re nearly there, but actually we can simplify this fraction because 16 and six are both even numbers. If we divide 16 by two, we get eight. And dividing six by two, we get three, so the fraction simplifies to eight over three. We could also convert our answer to a mixed number, as it’s currently a top heavy fraction. Three goes into eight twice with a remainder of two. So eight over three can be written as the mixed number two and two-thirds. Either of these formats would be acceptable for your answer.
Now I said we were going to consider two methods. So we’ve looked at the formal algebraic method, and the other method is to consider what we call a number machine. We can represent this equation using a number machine. We start with an input of 𝑥 multiplied by six to give six 𝑥, add four, and it gives an output of 20.
Solving an equation is just like working backwards through a number machine. So we start on the right with an output of 20 and work our way back, performing the opposite or inverse operations each time. The opposite or inverse of adding four is subtracting four, so first we subtract four from 20 to give 16. The opposite or inverse of multiplying by six is dividing by six, so then we divide 16 by six to give the fraction 16 over six. This is the same as the fraction we had at this stage of our working out in the formal algebraic method. So it could be simplified to eight over three or two and two-thirds in the same way.
Notice that we’ve performed exactly the same steps to solve this equation in both of our methods. First, we subtracted four, and then we’ve divided by six. The number machine gives more visual representation of what we’re doing, but the algebraic approach is a more formal one. In either case, it gives the same result. The solution to the equation six 𝑥 plus four is equal to 20 is 𝑥 is equal to eight over three or two and two-thirds.
Now it’s always sensible to check our answer when we can. So to do this, we can substitute the value of eight over three back into the left side of the equation. So six 𝑥 plus four becomes six multiplied by eight over three plus four.
Now to perform this multiplication, we can think of six as six over one. So now we have two fractions multiplied together. We can do some cross-cancelling to simplify this product, as six and three have a common factor of three. Six divided by three is two, and three divided by three is one. So the product simplifies to two over one multiplied by eight over one.
To multiply fractions, we just multiply the numerators together and multiply the denominators. So we have two multiplied by eight, which is 16, and one multiplied by one, which is one. But the fraction 16 over one is just equal to 16. Six 𝑥 plus four therefore becomes 16 plus four, which is equal to 20.
Looking back at the original equation, we can see that this is what six 𝑥 plus four is supposed to be equal to. So this check tells us that our value of eight over three or two and two-thirds is correct.