Amanda is making sandwiches for an event. The Venn diagram shows information about the sandwiches. There are 90 sandwiches made in total. 𝐴 is the number of sandwiches with brown bread. 𝐵 is the number of buttered sandwiches. Tom is choosing a sandwich at random. It has brown bread. Calculate the probability that the sandwich is buttered.
As there are 90 sandwiches made in total. We know that all the expressions inside the Venn diagram must total 90. The number of sandwiches with brown bread is everything inside the yellow circle. Also, the number of sandwiches that are buttered are inside the pink circle. The expression inside the intersection or overlap of the two circles — in this case 𝑥 multiplied by 𝑥 minus eight — is the number of sandwiches that are made from brown bread and are buttered. The expression outside the circles — in this case three 𝑥 minus six — is those sandwiches that are not made from brown bread or buttered.
As there were 90 sandwiches in total, we know that two 𝑥 minus seven plus 𝑥 multiplied by 𝑥 minus eight plus 15 plus three 𝑥 minus six equals 90. In order to simplify this equation, we firstly need to expand the bracket. 𝑥 multiplied by 𝑥 is equal to 𝑥 squared and 𝑥 multiplied by negative eight is equal to negative eight 𝑥. We are therefore left with two 𝑥 minus seven plus 𝑥 squared minus eight 𝑥 plus 15 plus three 𝑥 minus six is equal to 90. We now need to group or collect the like terms.
There is only one term with 𝑥 squared. Therefore, we can start with 𝑥 squared. Grouping the 𝑥 terms gives us negative three 𝑥 as two 𝑥 minus eight 𝑥 plus three 𝑥 is equal to negative three 𝑥. Grouping the constants gives us plus two. Negative seven plus 15 is equal to eight. Subtracting six from this gives us two. This means that 𝑥 squared minus three 𝑥 plus two is equal to 90. Subtracting 90 from both sides of this equation gives us 𝑥 squared minus three 𝑥 minus 88 is equal to zero as plus two minus 90 is equal to negative 88.
We now have a quadratic equation that we can solve by factorizing. The coefficient of 𝑥 squared is one. This means that the first term in both of the brackets will be 𝑥 as 𝑥 multiplied by 𝑥 gives us 𝑥 squared. The second term in both of the brackets will be integers or whole numbers. They need to have a product of negative 88 and a sum of negative three. This means the two numbers need to multiply to give us negative 88 and they need to add to give us negative three.
Two pairs of numbers that multiply to give us negative 88 are negative eight and 11 and negative 11 and eight. Remember that multiplying a negative number by a positive gives a negative answer. The second pair negative 11 and eight also has a sum of negative three as negative 11 plus eight equals negative three. This means that our two brackets are 𝑥 minus 11 and 𝑥 plus eight. We can write these two brackets in either order. But the eight must be positive and the 11 must be negative.
As the product of these two brackets is equal to nought, we have two possible solutions: either 𝑥 minus 11 is equal to zero or 𝑥 plus eight is equal to zero. Adding 11 to both sides of the first equation gives us 𝑥 is equal to 11. Subtracting eight from both sides of the second equation gives us 𝑥 is equal to negative eight. We therefore have two possible values of 𝑥: 𝑥 equals 11 or 𝑥 equals negative eight.
As the number of sandwiches must be positive, we can eliminate the answer 𝑥 equals negative eight. Therefore, 𝑥 must be equal to 11. We now need to substitute 𝑥 equals 11 into the three expressions from the Venn diagram. Substituting 𝑥 equals 11 into the expression two 𝑥 minus seven gives us two multiplied by 11 minus seven. Two multiplied by eleven is equal to 22. Subtracting seven from this gives us 15.
Substituting 𝑥 equals 11 into the expression 𝑥 multiplied by 𝑥 minus eight gives us 11 multiplied by eleven minus eight. 11 minus eight is equal to three and 11 multiplied by three is equal to 33. This means that the intersection between sandwiches with brown bread and buttered sandwiches is 33. Amanda made 33 sandwiches with brown bread that were buttered.
Finally, we need to substitute 𝑥 equals 11 into the expression three 𝑥 minus six. This gives us three multiplied by 11 minus six. Three multiplied by 11 is equal to 33 and subtracting six from this gives us 27. This means that there were 27 sandwiches that would not buttered and would not made with brown bread. We now have a completed Venn diagram with the values 15, 33, 15, and 27.
We were told that Tom is choosing a sandwich at random and that it has brown bread. This means that it must be inside circle A on the Venn diagram. The total number of sandwiches with brown bread is equal to 15 plus 33. Therefore, there are 48 sandwiches that were made with brown bread. We want to calculate the probability that this sandwich is buttered. How many of the 48 brown bread sandwiches are buttered? This is given by the intersection or the overlap of the two circles. Therefore, there are 33 sandwiches that are buttered and have brown bread. We can therefore say that the probability is 33 out of 48.
As the sandwich that Tom has chosen has got brown bread, the probability that it is buttered is 33 out of 48. We can simplify this fraction by dividing the numerator and denominator by three. Remember whatever you do to the top, you must do to the bottom. 33 divided by three is equal to 11 and 48 divided by three is equal to 16. This means that the probability in its simplest form is 11 out of 16 or eleven sixteenths.