🎯⭐ INTERACTIVE LESSON

Ratios, Proportions, and Percents

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Ratios, Proportions, and Percents - Complete Interactive Lesson

Part 1: Ratios & Rates

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?

Step	Work
Total parts	$2 + 3 + 5 = 10$
Blue fraction	$3/10$
Blue amount	$(3 / 10) (60) = 18$ liters

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	Calculation	Unit Rate
A	$8.50 ÷ 5$	$1.70/lb
B	$4.80 ÷ 3$	$1.60/lb

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?

Step	Work
Set up proportion	$\frac{\text{cats}}{\text{dogs}} = \frac{5}{3}$

Harder Ratio Problems 🎯

Ratio, Rate, or Proportion? 🔍

Classify each problem type.

Key Takeaways — Part 1

Concept	Formula	When to Use
Ratio $a:b$	Part $= \frac{a}{a+b} × T$

Part 2: Proportional Reasoning

Ratios, Proportions & Percentages

Part 2 of 7 — Percentages

Percent Basics

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

"What is 15% of 80?" → $0.15 \times 80 = 12$

Part 3: Percentages

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.

Part 4: Unit Conversion

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"

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

Part 5: Scale & Modeling

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 $k^3$

Part 6: Problem-Solving Workshop

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:

$0.60x + 0.20(10) = 0.40(x + 10)$

Part 7: Review & Applications

Ratios, Proportions & Percentages

Part 7 of 7 — Review & SAT Mixed Practice

Quick Reference

Topic	Key Formula
Ratio $a:b$ , total $T$	Part $= \frac{a}{a+b} \times T$

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

Step	Work
Gallons needed	$480 ÷ 32 = 15$ gallons
Cost	$15 × \$ 3.60 = $54$

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

"12 is what percent of 80?" →

\frac{12}{80} = 0.15 = 15\%

"12 is 15% of what?" →

12 = 0.15x

→

x = 80

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

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: $80 \times 1.20 = \$ 96$

15% discount on $200: $200 \times 0.85 = \$ 170$

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?

Step	Work
Multiplier for 30% off	$1 - 0.30 = 0.70$
Set up equation	$0.70x = 56$
Solve	$x = 56 ÷ 0.70 = \$ 80$

⚠️ 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?

Step	Work
After 25% rise	$200 × 1.25 = 250$
After 20% fall	$250 × 0.80 = 200$
Net change	$\frac{200 - 200}{200} = 0\%$

This time it happens to be zero — but that's because $1.25 × 0.80 = 1.00$ 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?

Component	Calculation	Amount
Meal	—	$42.00
Tax (8%)	$42 × 0.08$	$3.36
Tip (20%)	$42 × 0.20$	$8.40
Total	—	$53.76

With multiplier: $42 × (1 + 0.08 + 0.20) = 42 × 1.28 = \$ 53.76$

Worked Example 4

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

Step	Multiplier
40% markup	$1.40$
10% discount	$0.90$
Net	$1.40 × 0.90 = 1.26$

The net markup is 26%, not 30%.

Percent Applications 🎯

What Multiplier? 🔍

Choose the correct multiplier for each scenario.

Key Takeaways — Part 2

Scenario	Multiplier	Example
$p\%$ increase	$1 + p/100$	20% increase → $×1.20$
$p\%$ decrease	$1 - p/100$	15% off → $×0.85$
Find original	Divide by multiplier	$\text{sale} ÷ 0.75$
Successive changes	Multiply multipliers	$1.25 × 0.80 = 1.00$

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

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

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

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?

$x$	2	5	8	10
$y$	6	15	24	30

Check	$y/x$
$6/2$	$3$
$15/5$	$3$
$24/8$	$3$
$30/10$	$3$

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?

Step	Work
Workers × time = constant	$4 × 9 = 36$ worker-days
New equation	$6 × t = 36$
Solve	$t = 6$ days

Recognizing Variation on the SAT

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

Keyword	Type	Equation
"proportional to"	Direct	$y = kx$
"varies directly"	Direct	$y = kx$
"varies inversely"	Inverse	$xy = k$
"constant product"	Inverse	$xy = k$
"varies jointly with $x$ and $y$ "	Joint direct	$z = kxy$

Worked Example 3 — Joint Variation

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

Step	Work
Set up formula	$z = \frac{kx}{y^2}$
Find

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

Type	Equation	Constant	Graph
Direct	$y = kx$	$y/x = k$	Line through origin
Inverse	$y = k/x$	$xy = k$	Hyperbola
Joint	$z = kxy$	—	3D surface

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

3,600 seconds1 hour=

Common Conversions (SAT-relevant)

Given	Conversion
1 mile	5,280 feet
1 kilometer	1,000 meters
1 hour	60 minutes = 3,600 seconds
1 gallon	4 quarts
1 pound	16 ounces

SAT Unit Conversion Strategy

Write the starting quantity as a fraction
Multiply by conversion factors so unwanted units cancel
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?

Step	Work
Start	$5 \text{ gal/min}$
Gallons → quarts	$× \frac{4 \text{ qt}}{1 \text{ gal}} = 20 \text{ qt/min}$
Minutes → hours	$× \frac{60 \text{ min}}{1 \text{ hr}} = 1{,}200 \text{ qt/hr}$

Worked Example 2 — Area Conversion

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

Step	Work
Area in sq ft	$12 × 15 = 180 \text{ ft}^2$
Convert	$180 ÷ 9 = 20 \text{ yd}^2$

⚠️ 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.

Step	Work
Per hour	$480 ÷ 8 = 60$ widgets/hr
Per minute	$60 ÷ 60 = 1$ widget/min

Worked Example 4

A density is given as 2.7 g/cm³. Convert to kg/m³. (1 kg = 1000 g, 1 m = 100 cm)

Step	Work
Grams → kg	$÷ 1000$
cm³ → m³	$÷ (100)^3 = ÷ 1{,}000{,}000$
Combined	kg/m³

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

Conversion Chain Template

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

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

Situation	Key Idea
Single unit	Multiply by conversion fraction
Rate (two units)	Convert numerator AND denominator
Area units	Square the linear factor: ft² → yd² ÷ 9
Volume units	Cube the linear factor: ft³ → yd³ ÷ 27

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

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 $A \times k^2 = A_{\text{area}} \times 4$

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

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

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 cm², what is the area of the larger?

Step	Work
Scale factor	$k = 30/20 = 1.5$
Area factor	$k^2 = 2.25$
Larger area	$50 × 2.25 = 112.5$ cm²

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?

Step	Work
Scale factor	$k = 24$
Volume factor	$k^3 = 13{,}824$
Actual volume	$0.5 × 13{,}824 = 6{,}912$ mL $≈ 6.9$ liters

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?

Step	Work
Set up similar triangles	$\frac{\text{height}}{\text{shadow}} = \frac{6}{4}$
Apply to tree	$\frac{h}{20} = \frac{6}{4}$
Solve	$h = \frac{6 × 20}{4} = 30$ feet

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?

Step	Work
Actual distance	$6.5 × 40 = 260$ miles
Time	$260 ÷ 65 = 4$ hours

The Scale Factor Cheat Sheet

What scales?	Factor
Length, perimeter, height	$k$
Area, surface area	$k^2$
Volume, capacity, weight*	$k^3$

*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

Measurement	Scaling Factor	Example ( $k = 3$ )
Length	$k$	$×3$
Area	$k^2$	$×9$
Volume	$k^3$	$×27$
Angles	1 (unchanged)	Same

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

0.60x + 2 = 0.40x + 4

0.60 x + 2 = 0.40 x + 4

x = 10 \text{ liters}

"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?

Component	Volume	% Alcohol	Amount of Alcohol
Solution	40 mL	70%	28 mL
Water	$x$ mL	0%	0 mL
Mixture	$40 + x$ mL	50%	$0.50(40 + x)$ mL

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

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?

Step	Work
A's rate	$1/10$ per hour
B's rate	$1/15$ per hour
A does in 3 hrs	$3/10$ of the job
Remaining	$1 - 3/10 = 7/10$
Combined rate	$1/10 + 1/15 = 5/30 = 1/6$
Time for rest	$(7/10) ÷ (1/6) = 42/10 = 4.2$ hours

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?

Component	Volume	Juice
80% juice	$x$	$0.80x$
30% juice	12	$0.30(12) = 3.6$
Mixture	$x + 12$	$0.50(x + 12)$

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

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?

Step	Work
Fill rate	$+1/5$ tank/hr
Drain rate	$-1/8$ tank/hr
Net rate	$1/5 - 1/8 = 8/40 - 5/40 = 3/40$

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

Problem Type	Key Setup
Work (together)	$1/a + 1/b = 1/t$
Work (one starts early)	Find remaining work, then use combined rate
Fill & drain	Subtract drain rate: $1/a - 1/b$
Mixture	Amount₁ + Amount₂ = Amount_mix
Workers × time	Total work = workers × time (constant)

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

Dividing percent change by the new value instead of the old
Assuming successive percent increases/decreases cancel out
Adding times instead of rates in work problems
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?

Step	Work
Markup	$50 × 1.60 = \$ 80$
Sale discount	$80 × 0.75 = \$ 60$
Net multiplier	$1.60 × 0.75 = 1.20$ → 20% net markup

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?

Step	Work
Let parts	Boys $= 3k$ , Girls $= 5k$
After 4 boys join	$3k + 4 = 5k$
Solve	$4 = 2k$ → $k = 2$
Original total	$3(2) + 5(2) = 16$ students

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?

Step	Work
Original salary	$1.10x = 55{,}000$ → $x = \$ 50{,}000$
Bonus	$55{,}000 × 0.05 = \$ 2{,}750$
Total compensation	$55{,}000 + 2{,}750 = \$ 57{,}750$

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?

Step	Work
Real dimensions	$6 × 50 = 300$ cm, $8 × 50 = 400$ cm
Convert to meters	$3$ m × m

Key Takeaways — Full Topic Review

Category	Formula	Common Trap
Ratios	$\frac{a}{a+b} × T$	Forgetting to add parts
Proportions	$ad = bc$	Mismatched units
Percent change	$\frac{\Delta}{\text{old}} × 100$	Dividing by new value
Finding original	$\text{sale} ÷ (1 - r)$	Adding percent to sale price
Successive %	Multiply multipliers	Adding percentages
Direct variation	$y = kx$ , $y/x = k$	Confusing with inverse
Inverse variation	$xy = k$	Adding instead of multiplying
Scale factor	L → $k$ , A → $k^2$ , V → $k^3$	Using $k$ for area
Work problems	Add rates, not times	Adding times
Mixtures	Vol × Conc = Amount	Averaging concentrations

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

2.7 \times \frac{1 , 000 , 000}{1 , 000} = 2, 700

Ratios, Proportions, and Percents

Ratios, Proportions, and Percents - Complete Interactive Lesson

Part 1: Ratios & Rates

Ratios, Proportions & Percentages

Ratios

Rates

Proportions

Worked Example 1 — Three-Part Ratio

Worked Example 2 — Comparing Unit Rates

Ratio Problems with Unknowns

Worked Example 3

Key Takeaways — Part 1

Part 2: Proportional Reasoning

Ratios, Proportions & Percentages

Percent Basics

Part 3: Percentages

Ratios, Proportions & Percentages

Direct Variation: y=kxy = kxy=kx

Part 4: Unit Conversion

Ratios, Proportions & Percentages

Dimensional Analysis

Part 5: Scale & Modeling

Ratios, Proportions & Percentages

Scale Factor

Part 6: Problem-Solving Workshop

Ratios, Proportions & Percentages

Mixture Problems

Part 7: Review & Applications

Ratios, Proportions & Percentages

Quick Reference

Worked Example 4

Percent Change

Multiplier Method (Faster!)

Successive Percent Changes ⚠️

Worked Example 1 — Finding the Original

Worked Example 2 — Successive Changes

Tax, Tip, and Markup

Worked Example 3

Worked Example 4

Key Takeaways — Part 2

Inverse Variation: y=k/xy = k/xy=k/x

Joint Variation

SAT Application

Worked Example 1 — Direct Variation from a Table

Worked Example 2 — Inverse Variation Application

Recognizing Variation on the SAT

Worked Example 3 — Joint Variation

Worked Example 4 — Graph Clue

Key Takeaways — Part 3

Common Conversions (SAT-relevant)

SAT Unit Conversion Strategy

Worked Example 1 — Multi-Step Conversion

Worked Example 2 — Area Conversion

Conversions with Rates

Worked Example 3

Worked Example 4

Conversion Chain Template

Key Takeaways — Part 4

Similar Triangles

Map/Model Problems

SAT Application

Worked Example 1 — Area from Scale Factor

Worked Example 2 — Volume from Scale Factor

Similar Triangles on the SAT

Worked Example 3

Worked Example 4

The Scale Factor Cheat Sheet

Key Takeaways — Part 5

Work/Rate Problems

SAT Strategy for Rate Problems

Worked Example 1 — Mixture Table Method

Worked Example 2 — Work Problem with One Working Then Both

More Mixture Scenarios

Worked Example 3 — Mixing Two Concentrations

Worked Example 4 — Work Problem: Draining While Filling

Key Takeaways — Part 6

Common SAT Mistakes

Worked Example 1 — Multi-Concept

Worked Example 2 — Proportion with Ratio Shift

SAT Strategy Playbook

Direct Variation: $y = kx$

Inverse Variation: $y = k/x$