7.4 Ratios, Rates, and Staffing Comparisons

Comparing Numbers With Ratios and Rates

A ratio compares one quantity with another. In correctional scenarios, a ratio may compare inmates to officers, completed checks to required checks, incidents to days, or new arrivals to available beds. A rate compares a quantity to a unit such as per hour, per shift, per day, or per housing unit. A percentage is simply a ratio out of 100.

Simplify a ratio by dividing both numbers by their greatest common factor. A ratio of 60 inmates to 4 officers is 60:4; both divide by 4, giving 15:1 — fifteen inmates for each officer. Simplification does not by itself prove whether staffing is acceptable. Unless the prompt supplies a rule, the exam wants the calculation or the comparison, not a policy judgment from outside knowledge.

Officer-to-inmate staffing is the classic corrections ratio. If a 240-bed facility is fully occupied and policy in the scenario sets a 1:16 supervision ratio for the day shift, the required officers are 240 ÷ 16 = 15 officers. If only 12 are on the roster, the shortfall is 15 − 12 = 3 officers. Always answer from the ratio the scenario states; do not import a number from another agency.

Note the direction carefully: a 1:16 ratio means one officer per sixteen inmates, so you divide inmates by 16 to find officers — multiplying would give a nonsensical 3,840. A quick reasonableness glance (the answer should be far smaller than the inmate count) catches a reversed operation instantly.

Percentages, Averages, and Change Over Time

Percentages out of 100 show up as percentage of capacity and percent complete. If a 240-bed facility holds 312 inmates, occupancy is 312 ÷ 240 = 1.30 = 130 percent of capacity (overcrowded). If 12 of 48 scheduled rounds are complete, the completed share is 12 ÷ 48 = 0.25 = 25 percent; if the question instead asks how many remain, subtract: 48 − 12 = 36 rounds. Never give a percentage when the question asks for a count.

Percent change measures increase or decrease against the original amount: (new − old) ÷ old × 100. If a unit's monthly incidents rise from 20 to 25, the change is (25 − 20) ÷ 20 × 100 = 5 ÷ 20 × 100 = 25 percent increase. If they fall from 40 to 30, the change is (30 − 40) ÷ 40 × 100 = −10 ÷ 40 × 100 = 25 percent decrease. The denominator is always the original (old) value, not the new one.

Averages summarize a set: add the values and divide by how many there are. An officer who records 18, 20, and 22 checks over three shifts averages (18 + 20 + 22) ÷ 3 = 60 ÷ 3 = 20 checks per shift. The average does not mean every shift had exactly 20; it describes the set. Watch the divisor: it is the number of values, not any number that happens to appear in the problem. If a question gives four shifts but only three have data, decide from the wording whether the missing shift counts as zero (divide by 4) or is excluded (divide by 3) — the two readings give different averages, and the distractors will include both.

Fair comparison requires matching units

Rates compare fairly only when the unit is the same. If Unit A has 6 incidents in 3 days and Unit B has 8 incidents in 4 days, both are 2 incidents per day — equal. If one figure is per shift and another is per day, convert first or recognize that they are not directly comparable.

Ratios also catch the candidate who confuses count with rate. Unit A has 5 late rounds out of 50; Unit B has 4 late rounds out of 20. Unit A has more late rounds by count, but 5 ÷ 50 = 10 percent while 4 ÷ 20 = 20 percent, so Unit B has the higher rate. This count-versus-rate trap is common precisely because the unit with the larger raw number feels worse at a glance. Whenever two groups have different denominators, suspect that the question is testing rate rather than count.

Proportions for scaling

A proportion sets two ratios equal and is the tool for "if this, then how many" scaling. If the staffing standard is 3 officers for every 48 inmates, how many officers does a 160-inmate unit need? Set up 3/48 = x/160, cross-multiply to get 48x = 480, then x = 10 officers. Proportions also handle supply problems: if 4 cases of trays serve 96 inmates, then 1 case serves 24, and 300 inmates need 300 ÷ 24 = 12.5, rounded up to 13 cases because you cannot under-provide meals — follow any rounding rule the prompt states.

Keep correctional judgment modest: you can state which rate is higher, which total is lower, or which unit meets a stated rule — but do not declare a ratio a policy violation unless the question supplies that rule. The tested skill is accurate calculation plus disciplined rule application.