Ava and Emma want to know who is better at the new video game Ava got for her birthday. They are each going to play one level and whoever has the highest score wins. They receive 10 points for every enemy they jump on, 5 points for each berry they collect, and 30 points for every second left on the timer when they finish the level. If Ava jumps on 8 more enemies than Emma and collects 3 more berries, but finishes the level 4 seconds slower, what is the difference between their two scores?
Answer: The difference between their two scores is -25, meaning Ava's score is 25 points lower than Emma's score.
Step-by-step solution
Step 1: Define the scoring system
First, let's establish the formula for calculating a player's total score based on the given rules. Points are awarded for enemies jumped on, berries collected, and time remaining on the timer.
Step 2: Define variables for Emma's score
To compare their scores, we need to represent Emma's performance using variables. We'll use , , and for her enemies, berries, and time respectively.
Step 3: Write down Emma's score formula
Now we can write Emma's total score using the scoring system and the variables defined for her performance.
Step 4: Define variables for Ava's performance relative to Emma's
Next, we represent Ava's performance in terms of Emma's performance. Ava jumps on 8 more enemies, collects 3 more berries, and finishes 4 seconds slower, meaning she has 4 fewer seconds left on the timer.
Step 5: Write down Ava's score formula
Substitute Ava's performance values into the scoring formula to get an expression for Ava's total score.
Step 6: Expand Ava's score formula
Distribute the multipliers and simplify the terms in Ava's score expression. Combine the constant terms to further simplify.
Step 7: Calculate the difference in scores
To find the difference between their two scores, we subtract Emma's score from Ava's score. Observe how the variable terms cancel out, leaving only the constant difference.
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