If Jeff finds 5 grasshoppers, how many ladybugs and crickets does he find, knowing he finds 10 fewer grasshoppers than crickets and 5 fewer crickets than ladybugs?

Mathematics Questions

If Jeff finds 5 grasshoppers, how many ladybugs and crickets does he find, knowing he finds 10 fewer grasshoppers than crickets and 5 fewer crickets than ladybugs?

Short Answer

The solution involves defining variables for grasshoppers (x), crickets (y), and ladybugs (z), establishing two equations based on their relationships. By solving the equations with the given information, we find the counts: 5 grasshoppers, 15 crickets, and 20 ladybugs.

Step-by-Step Solution

Step 1: Define Variables

In this scenario, we need to establish variables for each type of insect:

x for grasshoppers
y for crickets
z for ladybugs

By defining these variables, we can easily express relationships between the different insect populations based on the information provided.

Step 2: Set Up Equations

From the conditions given, we can create two equations:

Equation I: The number of grasshoppers is 10 less than the number of crickets. This gives us x = y – 10.
Equation II: The number of crickets is 5 less than the number of ladybugs, resulting in y = z – 5.

These equations will guide us in finding the actual counts of each insect type.

Step 3: Solve the Equations

Now, to find the specific numbers:

We know from the problem that Jeff found 5 grasshoppers, so we substitute that into Equation I: 5 = y – 10. Solving this gives us y = 15 crickets.
Next, we substitute the value of y into Equation II: 15 = z – 5. This leads us to find z = 20 ladybugs.

Therefore, the final counts are 15 crickets and 20 ladybugs.

how many ladybugs and crickets…