Ratios and Proportions: TEAS Practice Question with Answer & Explanation

Understanding Proportions and Similar Triangles

To solve this problem, you must understand the concept of proportional reasoning or similar triangles. In the ATI TEAS 7 Math section, you will often encounter "shadow problems." These problems rely on the fact that since the sun is hitting both objects at the same angle at the same time, the ratio of the height of an object to the length of its shadow is constant.

The Set-Up

We can express the relationship as a proportion: $$\frac{\text{Height 1}}{\text{Shadow 1}} = \frac{\text{Height 2}}{\text{Shadow 2}}$$

From the problem, we have the following known values:

• Object 1 (Yardstick): Height = 3 feet, Shadow = 4 feet.
• Object 2 (Flagpole): Height = $x$ (our unknown), Shadow = 27 feet.

Solving the Equation

Set up the proportion: $$\frac{3}{4} = \frac{x}{27}$$

To solve for $x$, use cross-multiplication:

1. Multiply 3 by 27: $3 \times 27 = 81$
2. Multiply 4 by $x$: $4x$
3. Set them equal: $4x = 81$
4. Divide both sides by 4: $x = \frac{81}{4}$
5. $x = 20.25$

The height of the flagpole is 20.25 feet.

Analysis of Incorrect Answers

• 12.0 feet: This is an incorrect calculation likely resulting from multiplying 3 by 4 instead of setting up a ratio.
• 28.0 feet: This might occur if a student incorrectly adds or subtracts values rather than calculating the ratio.
• 30.5 feet: This is a distracter value that might appear if the division was performed incorrectly or if a "guess" was made based on the shadow being longer than the height.

TEAS Concept: Ratios and Proportions

On the TEAS, standard word problems often require converting a real-world scenario into a fraction. Remember: whatever unit or category you put on the top (numerator) of the first fraction must stay on the top of the second fraction. If you put "Height" on top for the yardstick, you must put "Height" on top for the flagpole.

Nursing Application

Why does a nursing student need to know this? Proportions are the foundation of dosage calculations. If a medication comes as 250mg per 5mL, and you need to give 1000mg, you are setting up a proportion exactly like this shadow problem to find the correct volume to administer. Finding the 'unknown' factor is a daily task in clinical practice.

Study Tip

When solving proportions, always perform a "sanity check." In this problem, the yardstick's shadow (4) is longer than its height (3). Therefore, you know the flagpole's shadow (27) must also be longer than its actual height. If your answer was larger than 27, you would immediately know you made a calculation error.