Modeling and Problem Solving (ACT Math)

What is Mathematical Modeling?

Mathematical modeling involves using math to represent real-world situations. On the ACT, this means:

• Translating word problems into equations
• Interpreting graphs and tables
• Applying math to practical scenarios
• Making predictions and drawing conclusions

Word Problem Strategy

The 4-Step Process

Step 1: Understand the Problem

• What are you asked to find?
• What information is given?
• Are there any constraints or conditions?

Step 2: Translate to Math

• Define variables
• Write equations or inequalities
• Identify which operation(s) to use

Step 3: Solve

• Apply appropriate mathematical techniques
• Work systematically
• Check your work as you go

Step 4: Verify and Interpret

• Does your answer make sense in context?
• Are units correct?
• Did you answer what was asked?

Common Word Problem Types

Distance, Rate, and Time

Formula: $d = rt$ (distance = rate × time)

Also useful:

• $r = \frac{d}{t}$
• $t = \frac{d}{r}$

Example: A car travels at 60 mph for 2.5 hours. How far does it go?

$d = 60 \times 2.5 = 150 \text{ miles}$

Example 2: Two cars leave the same point going opposite directions at 50 mph and 60 mph. How far apart after 3 hours?

Car 1: $d_1 = 50 \times 3 = 150$ miles
Car 2: $d_2 = 60 \times 3 = 180$ miles
Total distance apart: $150 + 180 = 330$ miles

Work Rate Problems

Formula: $\text{Work} = \text{Rate} \times \text{Time}$

Key: If someone completes a job in $n$ hours, their rate is $\frac{1}{n}$ job per hour

Example: Alice can paint a room in 6 hours. Bob can paint it in 4 hours. How long if they work together?

Alice's rate: $\frac{1}{6}$ room/hour
Bob's rate: $\frac{1}{4}$ room/hour
Combined rate: $\frac{1}{6} + \frac{1}{4} = \frac{2}{12} + \frac{3}{12} = \frac{5}{12}$ room/hour

Time together: $\frac{1}{\frac{5}{12}} = \frac{12}{5} = 2.4$ hours

Mixture Problems

Strategy: Set up equation based on the component you're tracking

Example: How many liters of 20% acid solution must be added to 10 liters of 50% acid solution to get 30% solution?

Let $x$ = liters of 20% solution

Equation (acid amounts): $0.20x + 0.50(10) = 0.30(x + 10)$ $0.20x + 5 = 0.30x + 3$ $2 = 0.10x$ $x = 20 \text{ liters}$

Age Problems

Strategy: Set up equation comparing ages at different times

Example: Sarah is currently 3 times as old as Tom. In 6 years, she'll be twice as old as Tom. How old is Tom now?

Let $t$ = Tom's current age

Now: Sarah is $3t$
In 6 years: Tom is $t + 6$, Sarah is $3t + 6$

Equation: $3t + 6 = 2(t + 6)$ $3t + 6 = 2t + 12$ $t = 6$

Tom is currently 6 years old.

Percent Increase/Decrease

Percent change: $\frac{\text{new} - \text{old}}{\text{old}} \times 100\%$

Successive changes: Must apply one at a time (they don't add!)

Example: A price increases 20%, then decreases 20%. Is it back to original?

Start with $100:

• After 20% increase: $100 \times 1.20 = 120$
• After 20% decrease: $120 \times 0.80 = 96$

No! It's now $96, not$100 (4% less than original)

Interpreting Graphs and Models

Linear Models

Form: $y = mx + b$

• $m$ = slope (rate of change)
• $b$ = y-intercept (starting value)

Example: A gym charges $30 initiation fee plus$15 per month.

Model: $C = 15m + 30$

• $C$ = total cost
• $m$ = number of months
• Slope (15) = monthly rate
• Intercept (30) = one-time fee

Questions:

• Cost for 8 months? $C = 15(8) + 30 = 150$
• How many months for $180?$180 = 15m + 30$→$m = 10$

Quadratic Models

Form: $y = ax^2 + bx + c$

Often models:

• Projectile motion: $h(t) = -16t^2 + v_0t + h_0$
• Area: $A = x(50 - x)$ for optimization
• Profit: $P = -2x^2 + 100x - 500$

Example: A ball is thrown with height $h(t) = -16t^2 + 64t + 5$ (in feet, $t$ in seconds)

Questions:

• Maximum height? Find vertex: $t = -\frac{b}{2a} = -\frac{64}{2(-16)} = 2$ seconds
  $h(2) = -16(4) + 64(2) + 5 = 69$ feet

• When does it hit ground? Set $h(t) = 0$ and solve

Exponential Models

Growth: $y = a(1 + r)^t$ where $r$ is growth rate
Decay: $y = a(1 - r)^t$ where $r$ is decay rate

Example: Population starts at 10,000 and grows 3% per year.

Model: $P(t) = 10000(1.03)^t$

After 5 years: $P(5) = 10000(1.03)^5 \approx 11,593$

Setting Up Equations from Descriptions

Key Phrases

Addition:

• sum, total, more than, increased by
• "5 more than $x$" → $x + 5$

Subtraction:

• difference, less than, decreased by, fewer
• "5 less than $x$" → $x - 5$
• "5 fewer than $x$" → $x - 5$
• CAREFUL: "5 subtracted from $x$" → $x - 5$ (NOT $5 - x$)

Multiplication:

• product, times, of (with percent/fraction)
• "20% of $x$" → $0.20x$
• "twice $x$" → $2x$

Division:

• quotient, per, ratio
• "$x$ divided by 5" → $\frac{x}{5}$
• "the ratio of $x$ to 5" → $\frac{x}{5}$

Equals:

• is, equals, is equal to, is the same as

Problem-Solving Strategies

Strategy 1: Plug In Answer Choices

When: Problem asks "which value" or gives numeric choices

How: Start with middle choice (often C), test if it works

Example: For what value of $x$ does $2x + 5 = 13$?
A) 2 B) 4 C) 6 D) 8 E) 10

Try C (6): $2(6) + 5 = 17$ (too big)
Try B (4): $2(4) + 5 = 13$ ✓

Strategy 2: Pick Numbers

When: Problem has variables and asks "which expression"

How: Choose simple numbers for variables, test each answer choice

Example: If $n$ is an even integer, which is always odd?

Pick $n = 4$:

• $n + 1 = 5$ (odd) ✓
• $2n = 8$ (even)
• $n^2 = 16$ (even)

Strategy 3: Draw a Diagram

When: Geometry or spatial reasoning problems

How: Sketch the situation, label what you know

Helps with: Triangle problems, distance problems, optimization

Strategy 4: Work Backwards

When: You know the end result and need to find the start

Example: After increasing a number by 20% and then subtracting 15, the result is 45. What was the original number?

Work backwards:

• Before subtracting 15: $45 + 15 = 60$
• Before 20% increase: $\frac{60}{1.20} = 50$

Original number: 50

Strategy 5: Create a Table or List

When: Pattern problems or multiple cases to track

How: Organize information systematically

Example: A bacteria population doubles every hour. If it starts at 100, what's the population after 4 hours?

| Hour | Population | |------|------------| | 0 | 100 | | 1 | 200 | | 2 | 400 | | 3 | 800 | | 4 | 1600 |

Answer: 1600

ACT Question Types

Type 1: Direct Translation

Strategy: Convert words to math step-by-step

Type 2: Rate Problems

Strategy: Identify $d = rt$ components, set up equation

Type 3: Optimization

Strategy:

• Set up equation for what you're maximizing/minimizing
• Often involves quadratic with vertex

Type 4: Graph Interpretation

Strategy:

• Identify what each axis represents
• Read slope as rate of change
• Use points to answer questions

Type 5: Multi-Step Problems

Strategy:

• Break into smaller steps
• Solve one part at a time
• Use first answer to get second

Common ACT Mistakes

❌ Solving for wrong variable — read what they ask for!
❌ Not checking answer in context — negative age? That's impossible!
❌ Rushing translation — "5 less than $x$" is $x - 5$, not $5 - x$
❌ Forgetting units — mixing hours and minutes, miles and feet
❌ Adding percents — 20% increase then 30% increase ≠ 50% increase!
❌ Not labeling variables clearly — leads to equation errors

Quick Tips for ACT

✓ Underline what you're asked to find — stay focused
✓ Define variables explicitly — write "$x$ = number of hours"
✓ Draw pictures for geometry and distance problems
✓ Check units — convert to same units before calculating
✓ Estimate first — helps eliminate wrong answers
✓ Use answer choices — can plug in to check
✓ Read ENTIRE problem before starting

Formula Reference

| Situation | Formula | |-----------|---------| | Distance | $d = rt$ | | Work Rate | $\text{Rate} = \frac{1}{\text{time to complete}}$ | | Combined Work | $\text{Rate}_1 + \text{Rate}_2 = \text{Rate}_{\text{together}}$ | | Percent Change | $\frac{\text{new} - \text{old}}{\text{old}} \times 100\%$ | | Linear Growth | $y = mx + b$ | | Exponential Growth | $y = a(1 + r)^t$ | | Projectile Height | $h(t) = -16t^2 + v_0t + h_0$ |

Practice Approach

1. Read twice — once for overall, once for details
2. Identify what you're finding — circle or underline it
3. List what you know — write down given information
4. Choose a strategy — translate, plug in, draw, etc.
5. Set up equation carefully — define variables
6. Solve systematically — show work
7. Check in context — does answer make sense?
8. Verify you answered the question — did they ask for $x$ or $2x + 5$?

Remember: Modeling problems test your ability to apply math to real situations. Stay organized, translate carefully, and always check if your answer makes sense!