Percentages · Ratios, Proportions, and Dosage

Mathematics · 3 min read

Percentages · Ratios, Proportions, and Dosage

Percent means per hundred. To change a percent to a decimal, divide by 100 (45% = 0.45); to change a decimal to a percent, multiply by 100 (0.072 = 7.2%). Every percent problem is a version of part = percent × whole.

The Three Percent Questions
Question	Method	Example
What is 30% of 80?	Multiply: 0.30 × 80	24
15 is what percent of 60?	Part ÷ whole × 100	15 ÷ 60 × 100 = 25%
20% of what number is 50?	Part ÷ percent	50 ÷ 0.20 = 250
Percent change	(new − original) ÷ original × 100	40 → 50: 10 ÷ 40 = 25% increase
Value after a change	Multiply by (1 ± rate)	$150 with 20% off: 150 × 0.80 = $120

Key Concept

Percent change always divides by the original value. A drop from 80 to 60 is a 25% decrease (20 ÷ 80), but a rise from 60 back to 80 is a 33.3% increase (20 ÷ 60).

Successive percentage changes multiply; they do not add. A 10% increase followed by a 10% decrease leaves 1.10 × 0.90 = 0.99 of the original, a net 1% decrease.

A ratio compares two quantities (3:5, 3/5, or 3 to 5) and is simplified like a fraction. A rate compares quantities with different units, such as mg per kg or mL per hour. A proportion states that two ratios are equal; solve it by cross-multiplying: 3/4 = x/20 gives 4x = 60, so x = 15.

Direct proportion: both quantities rise together. If 5 vials cost $15, 8 vials cost 8 × 3 = $24.
Inverse proportion: one rises as the other falls. If 4 aides finish a task in 12 hours, 6 aides need (4 × 12) ÷ 6 = 8 hours.
Scale: a scale of 1:100 means 1 unit on the drawing equals 100 units in reality.

Dosage by Proportion

Dosage problems are proportions in disguise. Set up have : have-in = want : x, keeping units matched on each side, or use the formula dose ordered ÷ dose on hand × quantity.

Worked Dosage Examples
Problem	Set-up	Answer
Order 0.375 mg; tablets are 0.25 mg	0.375 ÷ 0.25 × 1 tablet	1.5 tablets
Order 150 mg; suspension 50 mg per 5 mL	150 ÷ 50 × 5 mL	15 mL
Order 7.5 mg/kg for a 44-lb child	44 ÷ 2.2 = 20 kg; 20 × 7.5	150 mg
Order 500 mg; vial 250 mg/mL	500 ÷ 250 × 1 mL	2 mL
1,000 mL over 8 hours by pump	1,000 ÷ 8	125 mL/h
1,000 mL over 8 h, drop factor 15 gtt/mL	(1,000 × 15) ÷ (8 × 60)	31.25 → 31 gtt/min

Key Concept

Convert units before you set up the proportion. If the order is in grams and the label is in milligrams, change grams to milligrams first (1 g = 1,000 mg), then calculate.

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Percentages · Ratios, Proportions, and Dosage

The Three Percent Questions
Question	Method	Example
What is 30% of 80?	Multiply: 0.30 × 80	24
15 is what percent of 60?	Part ÷ whole × 100	15 ÷ 60 × 100 = 25%
20% of what number is 50?	Part ÷ percent	50 ÷ 0.20 = 250
Percent change	(new − original) ÷ original × 100	40 → 50: 10 ÷ 40 = 25% increase
Value after a change	Multiply by (1 ± rate)	$150 with 20% off: 150 × 0.80 = $120

Key Concept

Percent change always divides by the original value. A drop from 80 to 60 is a 25% decrease (20 ÷ 80), but a rise from 60 back to 80 is a 33.3% increase (20 ÷ 60).

Successive percentage changes multiply; they do not add. A 10% increase followed by a 10% decrease leaves 1.10 × 0.90 = 0.99 of the original, a net 1% decrease.

Direct proportion: both quantities rise together. If 5 vials cost $15, 8 vials cost 8 × 3 = $24.
Inverse proportion: one rises as the other falls. If 4 aides finish a task in 12 hours, 6 aides need (4 × 12) ÷ 6 = 8 hours.
Scale: a scale of 1:100 means 1 unit on the drawing equals 100 units in reality.

Dosage by Proportion

Dosage problems are proportions in disguise. Set up have : have-in = want : x, keeping units matched on each side, or use the formula dose ordered ÷ dose on hand × quantity.

Worked Dosage Examples
Problem	Set-up	Answer
Order 0.375 mg; tablets are 0.25 mg	0.375 ÷ 0.25 × 1 tablet	1.5 tablets
Order 150 mg; suspension 50 mg per 5 mL	150 ÷ 50 × 5 mL	15 mL
Order 7.5 mg/kg for a 44-lb child	44 ÷ 2.2 = 20 kg; 20 × 7.5	150 mg
Order 500 mg; vial 250 mg/mL	500 ÷ 250 × 1 mL	2 mL
1,000 mL over 8 hours by pump	1,000 ÷ 8	125 mL/h
1,000 mL over 8 h, drop factor 15 gtt/mL	(1,000 × 15) ÷ (8 × 60)	31.25 → 31 gtt/min

Key Concept

Convert units before you set up the proportion. If the order is in grams and the label is in milligrams, change grams to milligrams first (1 g = 1,000 mg), then calculate.

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