3.2 Ratios & Proportions

What Is a Ratio?

A ratio compares two (or more) quantities. There are two flavors:

• Part-to-part: compares one part of a group to another part. "8 nurses to 2 doctors" → 8:2 or 8/2 = 4/1.
• Part-to-whole: compares one part to the total. "8 nurses out of 10 staff" → 8/10 = 4/5.

Ratios can be written three ways: 3:2, 3/2, or "3 to 2." On the NEX you will usually see them as fractions or with a colon.

Example: A unit has 12 day-shift nurses and 18 night-shift nurses.

• Part-to-part, day : night = 12 : 18 = 2 : 3 (divide both by 6).
• Part-to-whole, day : total = 12 : 30 = 2 : 5.

Equivalent Ratios

Two ratios are equivalent if their cross-products are equal. You build equivalent ratios by multiplying (or dividing) both terms by the same nonzero number, just like fractions.

3 : 4 = 6 : 8 = 15 : 20 = 75 : 100.

To check whether 7/12 and 21/36 are equivalent, cross-multiply: 7 × 36 = 252 and 12 × 21 = 252. Equal, so yes.

Unit Rates

A rate is a ratio with different units (miles per hour, dollars per dose). A unit rate reduces the denominator to 1.

A patient receives 750 mL over 5 hours. What is the unit rate (mL/hr)? 750 ÷ 5 = 150 mL/hr.

A pharmacy charges $96 for 12 syringes. Unit price? 96 ÷ 12 = $8 per syringe.

Unit rates are how you compare prices and speeds: whichever has the smaller unit rate (per item, per hour) is the better buy or the faster rate.

Solving Proportions

A proportion is a statement that two ratios are equal: a/b = c/d. Solve an unknown by cross-multiplying: a × d = b × c, then isolate the variable.

Solve 3/4 = x/20. 3 × 20 = 4 × x → 60 = 4x → x = 15.

Solve 5/x = 15/42. 5 × 42 = 15 × x → 210 = 15x → x = 14.

Step-by-step procedure:

1. Write the proportion as two equal fractions.
2. Cross-multiply (top of one × bottom of the other).
3. Divide to isolate the unknown.

Setting Up Proportions From Word Problems

This is the core skill that feeds into dosage math. Translate the English into two equal ratios. Put matching units in matching positions.

A recipe uses 3 cups of flour for every 8 servings. How many cups for 20 servings? Set up: 3 cups / 8 servings = x cups / 20 servings. Cross-multiply: 3 × 20 = 8 × x → 60 = 8x → x = 7.5 cups.

The trick is to keep units in the same positions: "cups over servings = cups over servings," not "cups over servings = servings over cups."

Nursing-Style Proportion: Dose by Weight

A child weighs 18 kg. The ordered dose is 5 mg per kg per day. The daily dose is 5 mg/kg × 18 kg = 90 mg per day.

Set up a proportion to find how many 25-mg tablets to give per day: 90 mg / x tablets = 25 mg / 1 tablet → 90 = 25x → x = 3.6 tablets.

(Real practice would round and verify with a pharmacist, but the NEX tests the arithmetic.)

Solution Strength Proportion

A label reads "1 g of drug per 50 mL." How many mL contain 0.3 g? 1 g / 50 mL = 0.3 g / x mL → 1 × x = 50 × 0.3 → x = 15 mL.

Notice we kept grams on top on both sides and mL on the bottom on both sides. Same position rule.

Scale: Maps and Drawings

Scale problems are proportions in disguise.

A floor plan uses 1 cm = 4 m. A corridor measures 6.5 cm on the plan. Real length? 1 cm / 4 m = 6.5 cm / x m → x = 4 × 6.5 = 26 m.

Common Traps

1. Flipped units. If your first ratio is "mg/mL," the second ratio must also be "mg/mL," not "mL/mg."
2. Cross-multiplying incorrectly. Cross means across the equals sign: top-left × bottom-right = bottom-left × top-right.
3. Forgetting to reduce. 12:18 is fine, but 2:3 is the answer the NEX wants.
4. Mixing part-to-part and part-to-whole. "Out of 30 staff, 12 are day-shift" is part-to-whole (12/30); don't write it as 12:18 unless the question asks for day : night.

More Worked Examples

A saline drip runs at 150 mL/hr. How long will a 1,000-mL bag last? Unit rate already given: 150 mL/hr. Set up a proportion to find hours: 150 mL / 1 hr = 1,000 mL / x hr → 150x = 1,000 → x ≈ 6.67 hr (about 6 hr 40 min).

A medication order is 0.4 mg, and tablets come as 0.25 mg each. How many tablets should be given? 0.25 mg / 1 tablet = 0.4 mg / x tablets → 0.25x = 0.4 → x = 0.4/0.25 = 1.6 tablets. Real practice would round to a pharmacy-approved dose; the NEX tests the proportion.

A map scale is 1 in = 50 mi. Two cities are 3.4 in apart on the map. Actual distance? 1 in / 50 mi = 3.4 in / x mi → x = 50 × 3.4 = 170 mi.

A unit has 15 RNs and 10 LPNs. What is the ratio of RNs to total nursing staff? Total = 15 + 10 = 25. Ratio = 15 : 25 = 3 : 5. (Part-to-whole; reduce by dividing both by 5.)

Choosing the Whole

In "what percent" questions, the whole is whatever follows "of." Compare:

• "What percent of 80 is 20?" → whole = 80, part = 20 → 25%.
• "20 is 80% of what number?" → whole = unknown, part = 20, percent = 80. Set up 20 = 0.80 × x → x = 25. The wording flips the unknown, so read carefully before reaching for a formula.