A small bag contains four different flavors of chewy sweets: apple, orange, lemon, and strawberry. A sweet is chosen at random from the bag. The probability that it is lemon flavored is three fiftieths. Work out the probability that it is orange flavored. Assume that each individual sweet has the same probability of being picked.
The probability of an event occurring is found by dividing the number of ways that event can occur by the total number of possible outcomes. That means the probability of choosing a lemon sweet is found by dividing the number of lemon sweets there are by the total number of sweets in the bag. We have enough information in the table to help us form an equation representing this information. Let’s begin by finding the total number of sweets in the bag. We can find that by adding the number of apple sweets, the number of orange sweets, the number of lemon sweets, and the number of strawberry sweets. That’s 𝑥 plus six 𝑥 plus three plus two 𝑥 plus two.
Next, we’ll simplify this by collecting like terms: 𝑥 plus six 𝑥 plus two 𝑥 is nine 𝑥, and three plus two is five. So that tells us there are a total of nine 𝑥 plus five sweets in the bag. Three of these are lemon. Given that each sweet has the same probability of being picked, we can say the probability of choosing a lemon sweet is three out of the total number of sweets, that’s nine 𝑥 plus five. We have been given in the question the exact probability that the sweet is lemon flavored; it’s three fiftieths. This means we can say that the expression three over nine 𝑥 plus five must actually be equal to three fiftieths, the probability that it’s lemon flavored.
Three over nine 𝑥 plus five equals three over 50. Notice that the numerator, the top number, in each of these expressions is the same. Since the expressions are equal to each other, they’re the same; that tells us that they’re denominators must also be the same. Nine 𝑥 plus five must be the same as must be equal to 50. Let’s clear some space and then solve this equation. Remember, when we solve an equation, we’re looking to find the value or values of the unknown. Here, the unknown is 𝑥. To do this, we perform a series of inverse, that means opposite, operations.
Let’s begin by getting rid of the plus five. The opposite of adding five is subtracting five. And remember, we have to do that to both sides of the equation. Plus five minus five is zero, so on the left-hand side of this equation we have nine 𝑥. 50 minus five is 45, so that tells us that nine 𝑥 is equal to 45. Nine 𝑥 means nine multiplied by 𝑥. To get rid of that nine then, we do the opposite of multiplying; we divide both sides by nine. That then tells us that 𝑥 is equal to 45 divided by nine, which is five. So we’ve worked out that 𝑥 is equal to five. Now we can go back to our table and find the exact number of apple, orange, and strawberry sweets.
The number of apple sweets was 𝑥, so there are five apple sweets. The number of orange sweets is given by six 𝑥, so we can find the exact number of orange sweets by multiplying five by six; that’s 30. The number of strawberry sweets was two 𝑥 plus two, so we multiply five by two and then add two; that’s 12. To find the probability of choosing an orange sweet as required, we need to find the number of orange sweets there are and the total number of sweets in the bag. We just worked out that there are 30 orange sweets. To find the total number of sweets in the bag, we could either substitute our value of 𝑥 is equal to five into our expression for the total, that’s nine 𝑥 plus five, or we can add together the values we just found, that’s five plus 30 plus three plus 12; that’s 50.
You might have spotted that we didn’t actually need to do any calculations there. We formed an equation earlier saying that nine 𝑥 plus five is equal to 50. So we actually already knew that the total number of sweets was 50. This means that the probability of choosing an orange sweet is 30 over 50. We do need to give our answer in its simplest form. Here, both 30 and 50 share a factor of 10. 30 divided by 10 is three and 50 divided by 10 is five. And we can see the probability of choosing an orange sweet is three-fifths.