4.2 Solving Proportions & Cross-Multiplication

4.2 Solving Proportions & Cross-Multiplication

A proportion is an equation stating that two ratios or rates are equal. Proportions are among the most heavily tested mathematical structures on the ATI TEAS 7 exam because they represent the standard clinical method for calculating drug dosages, unit conversions, fluid volumes, and drip rates.

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The Structure of a Proportion

A proportion can be written in fractional form or colon form:

$$\frac{a}{b} = \frac{c}{d} \quad \text{or} \quad a : b = c : d$$

In colon notation ($a : b = c : d$):

• Extremes: The outer terms ($a$ and $d$).
• Means: The inner terms ($b$ and $c$).

The Fundamental Law of Proportions

The fundamental property governing all proportions states:

$$\text{Product of Extremes} = \text{Product of Means} \implies a \times d = b \times c$$

To verify whether two ratios form a true proportion, test whether their cross-products are equal. For instance, to verify if $\frac{4}{10}$ equals $\frac{10}{25}$:

$$4 \times 25 = 100 \quad \text{and} \quad 10 \times 10 = 100$$

Since both cross-products equal $100$, the equation $\frac{4}{10} = \frac{10}{25}$ is a valid proportion.

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Solving Proportions via Cross-Multiplication

When one value in a proportion is unknown (represented by the variable $x$), you can use cross-multiplication to solve for $x$.

The 3-Step Cross-Multiplication Method

1. Set Up the Equation: Write the two ratios as equivalent fractions with matching units in the numerators and matching units in the denominators: $$\frac{\text{Known Amount (Unit A)}}{\text{Known Volume (Unit B)}} = \frac{\text{Desired Dose (Unit A)}}{x \text{ (Unit B)}}$$
2. Cross-Multiply Diagonals: Multiply the numerator of the first fraction by the denominator of the second fraction, and equate it to the product of the remaining terms: $$a \cdot x = b \cdot c$$
3. Isolate the Variable ($x$): Divide both sides of the equation by the coefficient of $x$: $$x = \frac{b \cdot c}{a}$$

Worked Example 1: Solving a Standard Algebraic Proportion

Problem: Solve for $x$ in the proportional equation $\frac{3.5}{14} = \frac{x}{40}$.

• Step 1 (Cross-Multiply): Multiply diagonal terms: $$3.5 \times 40 = 14 \times x$$
• Step 2 (Simplify Product): Compute $3.5 \times 40 = 140$: $$140 = 14x$$
• Step 3 (Divide by Coefficient): Isolate $x$ by dividing both sides by $14$: $$x = \frac{140}{14} = 10$$
• Final Answer: $x = 10$.

Worked Example 2: Clinical Drug Dosage Calculation

Problem: A healthcare provider orders $250\text{ mg}$ of Amoxicillin oral suspension for a patient. The pharmacy dispenses Amoxicillin labeled $125\text{ mg / 5 mL}$. How many milliliters should the nurse administer to deliver the prescribed dose?

• Step 1 (Set Up Proportion): Align units consistently across numerators (mg) and denominators (mL): $$\frac{125\text{ mg}}{5\text{ mL}} = \frac{250\text{ mg}}{x\text{ mL}}$$
• Step 2 (Cross-Multiply): $$125 \cdot x = 5 \cdot 250$$ $$125x = 1,250$$
• Step 3 (Isolate $x$): $$x = \frac{1,250}{125} = 10\text{ mL}$$
• Final Answer: The nurse must administer $10\text{ mL}$.

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The 'Flipped Ratio' Trap: Maintaining Unit Alignment

The most frequent mistake students make on TEAS proportion problems is misaligning units across the equals sign. Consider the correct vs. incorrect setups for medication dosage:

CORRECT UNIT ALIGNMENT:
  Numerator:   [ Milligrams (mg) ]   =   [ Milligrams (mg) ]
  Denominator: [ Milliliters (mL) ]      [ Milliliters (mL) ]

INCORRECT (FLIPPED) ALIGNMENT:
  Numerator:   [ Milligrams (mg) ]   =   [ Milliliters (mL) ]  <-- WRONG!
  Denominator: [ Milliliters (mL) ]      [ Milligrams (mg) ]  <-- WRONG!

Golden Rule of Proportions: Whatever unit is in the top-left numerator MUST be the unit in the top-right numerator. Whatever unit is in the bottom-left denominator MUST be the unit in the bottom-right denominator.

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Direct vs. Inverse Proportions

On the TEAS exam, most ratio and proportion problems involve direct proportions:

• Direct Proportion: As one quantity increases, the second quantity increases proportionally ($y = kx$). For example, if $1\text{ tablet} = 50\text{ mg}$, then $2\text{ tablets} = 100\text{ mg}$, and $3\text{ tablets} = 150\text{ mg}$.
• Inverse Proportion: As one quantity increases, the second quantity decreases proportionally ($y = \frac{k}{x}$). An example is nurse staffing vs. time required to complete patient admissions: doubling the number of nurses cuts the required completion time in half.

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4-Function Calculator Shortcut for Proportions

When solving any proportion $\frac{a}{b} = \frac{c}{x}$ on the TEAS built-in calculator, use this fast 2-step calculation shortcut:

$$x = \frac{\text{Multiply the two known numbers on the diagonal}}{\text{Divide by the remaining known number}}$$

Calculator Keying Example for $\frac{125}{5} = \frac{250}{x}$:

1. Identify the full diagonal containing two known numbers: $5$ and $250$.
2. Key into calculator: 5 × 250 ÷ 125 =
3. Result: $10$.