Ratios, Proportions, and Percents

Ratios, Proportions & Percentages

Part 1 of 7 — Ratios and Rates

Ratios

A ratio compares two quantities: $a : b$ or $\frac{a}{b}$.

If the ratio of boys to girls is $3:5$ and there are 40 students total:

• Total parts = $3 + 5 = 8$
• Boys = $(3/8)(40) = 15$
• Girls = $(5/8)(40) = 25$

Rates

A rate is a ratio with units: miles/hour, dollars/item, people/year.

Unit Rate = rate per one unit. "$7.50 for 3 pounds" → $2.50 per pound.

Proportions

$\frac{a}{b} = \frac{c}{d} \implies ad = bc \quad \text{(cross multiply)}$

Example: If 3 widgets cost $14, how much do 7 widgets cost?

$\frac{3}{14} = \frac{7}{x} \implies 3x = 98 \implies x = \frac{98}{3} \approx \$32.67$

Worked Example 1 — Three-Part Ratio

In a mixture, red, blue, and yellow paint are in the ratio $2:3:5$. If the total is 60 liters, how much blue paint is there?

Worked Example 2 — Comparing Unit Rates

Store A sells 5 lb of apples for $8.50. Store B sells 3 lb for $4.80. Which is cheaper per pound?

Store B is cheaper by $0.10 per pound.

Ratios & Rates 🎯

Ratio Problems with Unknowns

Sometimes the SAT gives you a ratio and one part, not the total.

Worked Example 3

The ratio of cats to dogs at a shelter is $5:3$. If there are 24 dogs, how many cats are there?

Worked Example 4

A car averages 32 miles per gallon. Gas costs $3.60 per gallon. What is the fuel cost for a 480-mile trip?

Harder Ratio Problems 🎯

Ratio, Rate, or Proportion? 🔍

Classify each problem type.

Key Takeaways — Part 1

• Set up proportions with matching units on each side
• Three-part ratios: add all parts for the total
• Unit rates let you compare which deal is better

Ratios, Proportions & Percentages

Part 2 of 7 — Percentages

Percent Basics

$p\%$ means $\frac{p}{100}$.

• "What is 15% of 80?" → $0.15 \times 80 = 12$
• "12 is what percent of 80?" → $\frac{12}{80} = 0.15 = 15\%$
• "12 is 15% of what?" → $12 = 0.15x$ → $x = 80$

Percent Change

$\text{Percent change} = \frac{\text{New} - \text{Old}}{\text{Old}} \times 100\%$

Example: Price goes from $40 to $52: $\frac{52 - 40}{40} \times 100 = 30\%$ increase

Multiplier Method (Faster!)

• Increase of $p\%$: multiply by $1 + p/100$
• Decrease of $p\%$: multiply by $1 - p/100$

20% increase on $80: $80 \times 1.20 = 96$, i.e. $96

15% discount on $200: $200 \times 0.85 = 170$, i.e. $170

Successive Percent Changes ⚠️

A 20% increase followed by a 20% decrease is NOT back to the original!

$100 \times 1.20 = 120$, then $120 \times 0.80 = 96$ — that's a 4% net decrease.

Worked Example 1 — Finding the Original

After a 30% discount, a jacket costs $56. What was the original price?

⚠️ Common mistake: adding 30% to $56 gives $72.80, which is WRONG.

Worked Example 2 — Successive Changes

A stock rises 25% one year, then falls 20% the next. If it started at $200, what is its final value?

This time it happens to be zero — but that's because $1.25 \times 0.80 = 1.00$ exactly.

Percentages 🎯

Tax, Tip, and Markup

These are all "percent increase" problems.

Worked Example 3

A meal costs $42. Tax is 8% and tip is 20% (on the pre-tax amount). What is the total?

With multiplier: $42 \times (1 + 0.08 + 0.20) = 42 \times 1.28 = 53.76$, i.e. $53.76

Worked Example 4

A store marks up items 40% then offers a 10% "sale." What is the net markup?

The net markup is 26%, not 30%.

Percent Applications 🎯

What Multiplier? 🔍

Choose the correct multiplier for each scenario.

Key Takeaways — Part 2

• Percent change: always divide by the original
• Successive discounts: multiply multipliers, don't add percentages
• Tax + tip: both are increases on the base price
• "What was the original?" → divide by the multiplier, don't reverse-add

Ratios, Proportions & Percentages

Part 3 of 7 — Direct and Inverse Variation

Direct Variation: $y = kx$

"$y$ is directly proportional to $x$" means as $x$ doubles, $y$ doubles.

Constant of proportionality: $k = y/x$

Example: If $y = 12$ when $x = 4$, then $k = 3$ and $y = 3x$.

Inverse Variation: $y = k/x$

"$y$ is inversely proportional to $x$" means as $x$ doubles, $y$ halves.

Product is constant: $xy = k$

Example: If $y = 6$ when $x = 8$, then $k = 48$ and $y = 48/x$.

When $x = 12$: $y = 48/12 = 4$

Joint Variation

"$z$ varies directly with $x$ and inversely with $y$": $z = kx/y$

SAT Application

Speed and time for a fixed distance: $d = rt$, so $t = d/r$ — time is inversely proportional to rate.

If you double your speed, the trip takes half the time.

Worked Example 1 — Direct Variation from a Table

Does this table show direct variation?

Yes — $k = 3$ is constant, so $y = 3x$.

Worked Example 2 — Inverse Variation Application

It takes 4 painters 9 days to paint a building. How long for 6 painters?

Variation 🎯

Recognizing Variation on the SAT

The SAT often disguises variation problems. Here are keywords to watch for:

Worked Example 3 — Joint Variation

$z$ varies directly with $x$ and inversely with $y^2$. If $z = 8$ when $x = 4$ and $y = 1$, find $z$ when $x = 6$ and $y = 2$.

Worked Example 4 — Graph Clue

The graph of $y$ vs $x$ passes through the origin and is a straight line. What type of variation?

Direct variation — $y = kx$ always passes through $(0, 0)$ and has slope $k$.

If the graph is a hyperbola ($y = k/x$), it's inverse variation.

Variation Applications 🎯

Direct or Inverse? 🔍

Classify each relationship.

Key Takeaways — Part 3

• Find $k$ from a known pair, then use it for all other questions
• "Proportional to" on the SAT usually means direct variation
• Inverse variation: product stays constant as one goes up, the other goes down
• Watch for $y = kx^2$ or $y = k\sqrt{x}$ — these are NOT simple direct/inverse

Ratios, Proportions & Percentages

Part 4 of 7 — Unit Conversions

Dimensional Analysis

Convert units by multiplying by fractions equal to 1:

"Convert 30 mph to feet per second"

$30 \frac{\text{miles}}{\text{hour}} \times \frac{5,280 \text{ feet}}{1 \text{ mile}} \times \frac{1 \text{ hour}}{3,600 \text{ seconds}} = 44 \text{ ft/s}$

Common Conversions (SAT-relevant)

SAT Unit Conversion Strategy

1. Write the starting quantity as a fraction
2. Multiply by conversion factors so unwanted units cancel
3. Compute the result

The SAT provides conversion factors in the problem — you don't need to memorize them. Focus on the METHOD of canceling units.

Worked Example 1 — Multi-Step Conversion

A pump moves water at 5 gallons per minute. How many quarts per hour is that?

Worked Example 2 — Area Conversion

A room is 12 feet by 15 feet. What is the area in square yards? (1 yard = 3 feet)

⚠️ For area, you divide by $3^2 = 9$ (not by 3). For volume, divide by $3^3 = 27$.

Unit Conversions 🎯

Conversions with Rates

When converting rates, both the numerator and denominator units may change.

Worked Example 3

A factory produces 480 widgets per 8-hour shift. Express this in widgets per minute.

Worked Example 4

A density is given as 2.7 $g/cm^{3}$. Convert to $kg/m^{3}$. (1 kg = 1000 g, 1 m = 100 cm)

The numerator and denominator conversions partially cancel — a common SAT shortcut.

Conversion Chain Template

$\text{Start} \times \frac{\text{new unit}}{\text{old unit}} \times \frac{\text{new unit}}{\text{old unit}} = \text{Result}$

Write every step with units. If units don't cancel correctly, something is flipped.

Harder Conversions 🎯

Which Conversion Factor? 🔍

Pick the correct factor to go from the starting unit to the target unit.

Key Takeaways — Part 4

• Write units at every step — if they don't cancel, something is wrong
• SAT always provides conversion factors; focus on the method
• Going from bigger to smaller units → multiply; smaller to bigger → divide
• For area/volume: square or cube the conversion factor

Ratios, Proportions & Percentages

Part 5 of 7 — Scale Factors and Similar Figures

Scale Factor

If two figures are similar with a scale factor of $k$:

• Lengths scale by $k$
• Areas scale by $k^2$
• Volumes scale by $k^3$

Similar Triangles

Two triangles are similar if they have the same angles (AA similarity).

If triangle $A$ has sides 3, 4, 5 and triangle $B$ has a side of 6 corresponding to 3:

• Scale factor $k = 6/3 = 2$
• Other sides of $B$: $4 \times 2 = 8$ and $5 \times 2 = 10$
• Area of $B$ = Area of $A \times k^2 = A_{\text{area}} \times 4$

Map/Model Problems

"On a map, 1 inch = 25 miles. Two cities are 3.5 inches apart."

Distance $= 3.5 \times 25 = 87.5$ miles.

SAT Application

Scale factors appear in:

• Similar triangle problems
• Map and blueprint questions
• Geometry problems with dilations

Worked Example 1 — Area from Scale Factor

Two similar pentagons have perimeters of 20 cm and 30 cm. If the smaller has an area of 50 $cm^{2}$, what is the area of the larger?

Worked Example 2 — Volume from Scale Factor

A model car is built at 1:24 scale. If the model holds 0.5 mL of fuel in its tank, what does the actual tank hold?

Scale Factors 🎯

Similar Triangles on the SAT

The SAT loves problems where you must first identify similar triangles, then set up a proportion.

Worked Example 3

A 6-foot person casts a 4-foot shadow. A tree next to them casts a 20-foot shadow. How tall is the tree?

Worked Example 4

On a map, the scale is 1 inch : 40 miles. Two cities are 6.5 inches apart on the map. A car travels at 65 mph. How long is the drive?

The Scale Factor Cheat Sheet

*Weight scales as $k^3$ when density is the same.

Applied Scale Problems 🎯

What Power of $k$? 🔍

For each quantity, decide whether it scales by $k$, $k^2$, or $k^3$.

Key Takeaways — Part 5

• Find $k$ by dividing corresponding lengths
• Shadow problems → similar triangles → set up proportion
• Map problems → multiply map distance by scale
• The SAT frequently combines scale factors with other ratio concepts

Ratios, Proportions & Percentages

Part 6 of 7 — Mixture and Work Problems

Mixture Problems

"How many liters of 60% acid solution must be mixed with 10 liters of 20% acid to get a 40% solution?"

Let $x$ = liters of 60% solution:

$0.60x + 0.20(10) = 0.40(x + 10)$ $0.60x + 2 = 0.40x + 4$ $0.20x = 2$ $x = 10 \text{ liters}$

Work/Rate Problems

"Pipe A fills a tank in 6 hours, Pipe B in 4 hours. Together?"

• Rate A: $1/6$ tank/hour
• Rate B: $1/4$ tank/hour
• Combined: $1/6 + 1/4 = 2/12 + 3/12 = 5/12$ tank/hour
• Time: $12/5 = 2.4$ hours

SAT Strategy for Rate Problems

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

Add rates when working together. The combined rate is always faster than either individual rate.

Worked Example 1 — Mixture Table Method

A chemist has 40 mL of 70% alcohol and wants to dilute it to 50% alcohol by adding water. How much water?

$28 = 0.50(40 + x) \implies 28 = 20 + 0.5x \implies x = 16 \text{ mL}$

Worked Example 2 — Work Problem with One Working Then Both

Machine A takes 10 hours alone. Machine B takes 15 hours alone. If A works for 3 hours, then both work together, how much longer until done?

Mixtures & Work 🎯

More Mixture Scenarios

Worked Example 3 — Mixing Two Concentrations

How many liters of 80% juice must be mixed with 12 liters of 30% juice to get 50% juice?

$0.80x + 3.6 = 0.50x + 6$ $0.30x = 2.4$ $x = 8 \text{ liters}$

Worked Example 4 — Work Problem: Draining While Filling

A tap fills a tank in 5 hours. A drain empties it in 8 hours. If both are open, how long to fill?

The drain slows down the filling but doesn't stop it (fill rate > drain rate).

Advanced Rate Problems 🎯

Set Up the Equation 🔍

For each scenario, choose the correct equation setup.

Key Takeaways — Part 6

• Never add times — always convert to rates first
• Use a table for mixture problems: Volume × Concentration = Amount
• For fill-and-drain: if drain rate > fill rate, the tank never fills
• Combined time is always less than the fastest individual time

Ratios, Proportions & Percentages

Part 7 of 7 — Review & SAT Mixed Practice

Quick Reference

Common SAT Mistakes

1. Dividing percent change by the new value instead of the old
2. Assuming successive percent increases/decreases cancel out
3. Adding times instead of rates in work problems
4. Forgetting that the ratio $a:b$ has $a + b$ total parts, not $a \times b$

Worked Example 1 — Multi-Concept

A store buys items for $50 and marks them up 60%. During a sale, they offer 25% off. What is the sale price?

Worked Example 2 — Proportion with Ratio Shift

In a class, boys to girls is $3:5$. If 4 more boys join, the ratio becomes $1:1$. How many students were originally in the class?

Mixed Review 🎯

SAT Strategy Playbook

Worked Example 3 — Working Backwards

After a 10% raise, Maya earns $55,000. Then she gets a 5% bonus on the new salary. What was her original salary, and what is the bonus?

Worked Example 4 — Scale + Percent

A blueprint uses a 1:50 scale. A room on the blueprint is 6 cm × 8 cm. The builder adds 20% to the area for a patio. What is the total real area?

Top 5 SAT Strategies for This Topic

1. Is it direct or inverse? — If one goes up while the other goes down, it's inverse.
2. Use multipliers — Never add/subtract percentages directly.
3. Draw a table — For mixtures, lay out Volume × Concentration = Amount.
4. Check with easy numbers — Substitute $100$ to verify percent problems.
5. Units must match — In proportions, keep the same units on each side.

SAT-Level Challenge 🎯

Which Strategy? 🔍

For each problem, choose the best approach.

Key Takeaways — Full Topic Review

Final SAT Tips:

• Expect 4-6 ratio/proportion/percent questions per test
• Always write units to catch dimensional errors
• Use multipliers for percent problems — never add/subtract percentages directly
• For work problems: Rate × Time = Work, add rates when working together