8.2 Money Totals, Percentages, and Averages

Adding and Subtracting Money Totals

Police Problem Solving math questions frequently ask for a running dollar total - the combined value of stolen property, the total of several fines, or the amount of loss remaining after partial recovery. Every one of these is ordinary decimal addition or subtraction, and the only real risk is misaligning the decimal point between dollars and cents.

The safest paper method is to write every amount with exactly two decimal places, lining up the decimal points in a vertical column before adding, exactly the way whole numbers are lined up by place value.

Adding these column by column: cents first, 00 plus 50 plus 75 equals 125 cents, which is $1.25 - carry the extra dollar into the dollars column. Dollars: 420 plus 135 plus 89 equals 644, plus the carried dollar equals 645. Combined, the total stolen value is $645.25.

A follow-up question often asks for the loss remaining after partial recovery, which is simple subtraction from that same total. If police recover $210.00 worth of the stolen items, the remaining loss is $645.25 minus $210.00, or $435.25 - subtract cents first (25 minus 00 equals 25), then dollars (645 minus 210 equals 435), giving $435.25.

Percentages: Of, Off, and Remaining

'Percent' literally means 'per hundred,' so finding a percent of a number is the same operation as multiplying that number by a fraction with 100 on the bottom - or, equivalently, by the decimal form of the percent. Converting a percent to a decimal is always a matter of moving the decimal point two places to the left (25% becomes 0.25), and multiplying by that decimal gives the 'percent of' answer directly.

Worked example: a community recorded 180 reported incidents last year, and this year's total dropped by 20%. First find 20% of 180: the fast method finds 10% first (180 divided by 10 equals 18), then doubles it (18 x 2 equals 36). That 36 is the decrease, so this year's total is 180 minus 36, or 144 incidents.

'Percent remaining' problems add one extra step that is easy to skip under time pressure: after finding a percent used or spent, you must subtract that percent from the whole before answering, rather than reporting the percent-used figure as if it were the final answer. A patrol vehicle starts a shift with a full 60-liter fuel tank; a gauge check partway through the shift shows only 35% of the tank remains. The percent already used is 100% minus 35%, or 65%, and the liters used is 65% of 60 liters: 0.65 x 60. Breaking that down, 0.5 x 60 equals 30 and 0.15 x 60 equals 9, so 30 plus 9 equals 39 liters have been used, leaving 21 liters (60 minus 39) in the tank.

Mean Averages in Patrol and Report Data

A mean average - the type of average tested on the SSPO - is always found the same two-step way regardless of what is being averaged: add up every value to get a total, then divide that total by however many values were added. Skipping straight to division without first finding the full total is the single most common error on average questions done under time pressure.

Worked example: an officer's call volume over a five-day work week was 6, 9, 4, 7, and 9 calls per day. Step one, add all five values: 6 plus 9 is 15, 15 plus 4 is 19, 19 plus 7 is 26, 26 plus 9 is 35. Step two, divide the total by the number of values: 35 divided by 5 equals 7. The average is 7 calls per day.

A second worked example applies the same two steps to dollar amounts rather than counts, which is exactly how the exam might disguise the same skill inside a different scenario. Four traffic citations issued during a shift carried fines of $85, $120, $95, and $100. Adding: 85 plus 120 is 205, 205 plus 95 is 300, 300 plus 100 is 400. Dividing by the number of citations: 400 divided by 4 equals 100. The average citation amount was $100.

Averages, percentages, and money totals frequently combine inside a single multi-part scenario, so it helps to keep three habits fixed regardless of which combination appears: always find the full total before dividing for an average, always convert a percent to its decimal form before multiplying, and always line up decimal points carefully whenever dollar amounts are added or subtracted by hand. Because a fraction, a decimal, and a percent are three different ways of writing the exact same value, choosing whichever form makes the arithmetic cleanest for a specific problem - a fraction when the numbers divide evenly, a decimal when adding several values together - is a legitimate shortcut, not a different rule to memorize.