2.2 Percentage, Ratio, and Average Word Problems

Percentage, Ratio, and Average Word Problems

Law enforcement isn't just about single incidents; it involves analyzing trends, managing resources, and tracking performance over time. To do this, police departments heavily utilize percentages, ratios, and averages. On the NPOST, you will be expected to read a scenario, determine which mathematical concept applies, and perform the calculation accurately.

This section covers the formulas and methodologies required to master these common word problems.

Percentages

A percentage represents a portion of 100. Calculating percentages is crucial for determining crime reduction goals, budget allocations, and demographic statistics.

Finding a Percentage of a Total

To calculate a percentage of a number, convert the percentage to a decimal (by moving the decimal point two places to the left) and multiply it by the total.

Scenario 1: Budget Allocation A police department has a total annual training budget of $150,000. If 15% of this budget must be allocated specifically to de-escalation training, how much money is designated for this purpose?

Step-by-Step Solution:

1. Convert 15% to a decimal: 15 ÷ 100 = 0.15.
2. Set up the multiplication: 150000 x 0.15.
3. Ignore the decimal initially: 150000 x 15.
   • 150000 x 5 = 750000
   • 150000 x 10 = 1500000
   • 750000 + 1500000 = 2250000
4. Place the decimal point two spots from the right (because 0.15 has two decimal places): 22500.00. Answer: $22,500 is allocated for de-escalation training.

Calculating Percent Increase or Decrease

Police departments constantly track whether crime rates are going up or down. The formula for percent change is: Percent Change = (Difference between Old and New Value / Old Value) x 100

Scenario 2: Crime Reduction Last year, there were 40 reported auto thefts in the downtown district. This year, due to increased patrols, there were only 32 reported auto thefts. What is the percentage decrease in auto thefts?

Step-by-Step Solution:

1. Find the difference between the old and new value: 40 - 32 = 8.
2. Divide the difference by the original (old) value: 8 ÷ 40.
3. Perform the division: 8 ÷ 40 = 0.2. (Since 40 doesn't go into 8, add a decimal point and a zero to make it 80. 40 goes into 80 twice).
4. Multiply by 100 to get the percentage: 0.2 x 100 = 20. Answer: There was a 20% decrease in auto thefts.

Ratios

A ratio compares the size or quantity of two different things. Ratios are often used to express staffing levels, such as the number of officers compared to the civilian population, or the number of supervisors to patrol officers.

Ratios can be written using a colon (e.g., 1:5), as a fraction (1/5), or with the word "to" (1 to 5).

Solving Ratio Proportions

If you know a ratio and one part of a new total, you can find the missing number using equivalent fractions.

Scenario 3: Staffing Ratios A city maintains a standard ratio of 3 police officers for every 1,000 residents (3:1000). If a newly incorporated suburb has a population of 15,000 residents, how many officers should be assigned there to maintain the standard ratio?

Step-by-Step Solution:

1. Set up the proportion as fractions: 3 officers / 1000 residents = X officers / 15000 residents.
2. To solve for X, look at the relationship between the denominators. How do you get from 1000 to 15000? You multiply by 15.
3. Therefore, multiply the numerator by the same amount: 3 x 15 = 45. Answer: 45 officers should be assigned.

Scenario 4: Distributing by Ratio A task force of 24 investigators is being split between the narcotics division and the vice division in a 5:3 ratio. How many investigators will be assigned to the narcotics division?

Step-by-Step Solution:

1. Add the parts of the ratio together to find the total number of "parts" in the group: 5 + 3 = 8 parts total.
2. Divide the total number of investigators by the total number of parts to find the value of one part: 24 ÷ 8 = 3 investigators per part.
3. Multiply the value of one part by the number of parts assigned to the narcotics division (which is 5): 3 x 5 = 15. Answer: 15 investigators will go to the narcotics division.

Averages

The average, or arithmetic mean, is a single number that represents the middle value of a data set. Averages are heavily used to track officer performance, response times, and daily call volumes.

The formula is simple: Average = Sum of All Values / Total Number of Values.

Calculating Basic Averages

Scenario 5: Average Response Time A dispatcher tracks the response times for Priority 1 calls during a shift. The times (in minutes) are: 6, 4, 8, 5, and 7. What is the average response time for these calls?

Step-by-Step Solution:

1. Add all the values together: 6 + 4 + 8 + 5 + 7 = 30.
2. Count the number of values in the set. There are 5 calls.
3. Divide the sum by the number of values: 30 ÷ 5 = 6. Answer: The average response time is 6 minutes.

Finding a Missing Value Using an Average

Sometimes you are given the average and need to find a missing data point.

Scenario 6: Reaching a Target Average An officer has written citations for the past four days: 12, 9, 15, and 10 citations. The department expects an average of 12 citations per day over a five-day workweek. How many citations must the officer write on the fifth day to meet the exact average of 12?

Step-by-Step Solution:

1. If the average for 5 days must be 12, the total number of citations for all 5 days combined must be: 5 days x 12 citations/day = 60 total citations.
2. Add the citations from the first four days: 12 + 9 + 15 + 10 = 46.
3. Subtract the current total from the required total to find the missing day: 60 - 46 = 14. Answer: The officer must write 14 citations on the fifth day.

Mastering percentages, ratios, and averages requires careful reading to ensure you apply the right formula. Always double-check your work, as these questions frequently offer "distractor" multiple-choice answers that match common math errors.