Question

Jerry's average (arithmetic mean) score on the first 3 of 4 tests is $85.$ If Jerry wants to raise his average by 2 points, what score must he earn on the fourth test? A. $$87$$ B. $$89$$ C. $$90$$ D. $$93$$ E. $$95$$

Short answer

Jerry needs to score 93 on the fourth test.

Step-by-step solution

Understand the given information

Jerry's average score for the first 3 out of 4 tests is 85. This means the sum of the scores of the first 3 tests divided by 3 is equal to 85.

Calculate the total score for the first 3 tests

The average score is given by the formula: $$\text{Average}=\frac{\text{Sum of scores}}{\text{Number of tests}}$$ Hence, the sum of the first 3 test scores is: $$85\times 3=255$$

Determine the required average decision

Jerry wants to raise his average by 2 points, making the new average: $$85+2=87$$

Set up the equation for the required score on the fourth test

For the new average score including the fourth test: $$\text{New Average}=87=\frac{\text{Sum of all 4 scores}}{4}$$ The sum of all 4 scores will thus be: $$87\times 4=348$$

Calculate the score needed on the fourth test

Since the sum of the first 3 test scores is 255, the score needed on the fourth test can be found by subtracting this from the total sum needed: $$348-255=93$$

Verify the solution

To verify, the sum of the scores 255 + 93 = 348. Dividing by 4 confirms the new average: $$\frac{348}{4}=87$$

Key concepts

arithmetic mean

The arithmetic mean, often just called the average, is a measure of central tendency. It is calculated by summing all the values in a dataset and then dividing by the number of values. For example, in the exercise with Jerry's test scores, his average score is given by the formula: $\text{Average}=\frac{\text{Sum of scores}}{\text{Number of tests}}$. This tells us that the sum of Jerry's first three test scores divided by 3 is equal to 85. This concept is key to understanding the subsequent calculations in the exercise.

test score calculations

To find an unknown test score, it's necessary to first understand the given test scores and their average. In Jerry's exercise, we know his average score for the first three tests, which is 85. Therefore, to find the sum of these scores, we multiply the average by the number of tests: $85\times 3=255$. This gives us the total score for the first three tests. Knowing this total is crucial because it allows us to set up and solve for the score needed on the fourth test to achieve a new average.

average increase requirements

When a student wants to increase their average score, they need to know how much additional score is required on subsequent tests. In Jerry's case, he wants to raise his average by 2 points from 85 to 87. To determine what score he needs on the fourth test, we first recognize that the new average of 87 must include all four test scores. This can be represented with the equation: $\text{New Average}=87=\frac{\text{Sum of all 4 scores}}{4}$. Multiplying the new average by 4 gives us the total sum required for the four tests: $87\times 4=348$. Knowing this, we can now find the specific score Jerry needs on the fourth test by subtracting the sum of the first three test scores from the total required score.

math equations

Math equations help us solve for unknown values by establishing relationships between known and unknown variables. In Jerry’s test score problem, once we set up the equation for the required score on the fourth test, we can isolate the unknown variable. Starting with the total sum needed for all four tests: $87\times 4=348$. We then subtract the sum of the first three test scores: $348-255=93$. This tells us that Jerry needs to score 93 on his fourth test to achieve his desired average. Therefore, math equations are a powerful tool in finding solutions to such problems by using logical steps and operations.