🎯⭐ 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?

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

Ratios & Rates 🎯

Key Takeaways — Part 1

Ratio $a:b$ means the total has $a + b$ parts
Unit rate: divide to find the "per one" rate
Cross multiplication: $a/b = c/d$ →

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 $= \frac{a}{a+b} \times T$

Set up proportions with matching units: miles/hours = miles/hours

"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.

Percentages 🎯

Key Takeaways — Part 2

Percent change: divide the change by the original (not the new) value
Multiplier method is fastest: increase → multiply by $(1 + r)$ , decrease → $(1 - r)$
Successive percent changes: multiply the multipliers together
A% increase then A% decrease ≠ original (common SAT trap)

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.

Key Takeaways — Part 3

Direct: $y = kx$ (ratio is constant: $y/x = k$ )
Inverse: $y = k/x$ (product is constant: $xy = k$ )
"Proportional" on SAT usually means direct variation ( $y = kx$ )
Check with a known pair to find $k$ , then use $k$ for the new scenario

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.

Unit Conversions 🎯

Key Takeaways — Part 4

Dimensional analysis: multiply by fractions equal to 1, cancel units
Set up so unwanted units appear in numerator AND denominator (they cancel)
The SAT will give you all needed conversion factors — focus on the method
Write units at every step to avoid errors

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

Scale Factors 🎯

Key Takeaways — Part 5

Lengths → $k$ , Areas → $k^2$ , Volumes → $k^3$
Find the scale factor by comparing corresponding lengths
Map/model problems: set up a proportion with the scale
This length/area/volume relationship is a frequent SAT topic

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.

Mixtures & Work 🎯

Key Takeaways — Part 6

Mixtures: set up equation where (amount of substance) is equal on both sides
Work problems: add the rates (not the times), then find time from combined rate
Rate × Time = Work (or Amount)
For work problems: find individual rates, add them, then use $time = work/rate$

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 $a \times b$

Mixed Review 🎯

Key Takeaways — Part 7

Ratios, proportions, and percentages are core SAT topics — expect 4-6 questions per test
Percent change: always divide by the original
"Original price" problems: work backwards using the multiplier
Workers × time = constant (inverse variation) is a classic SAT setup

Ratios, Proportions, and Percents

Percent Change

Multiplier Method (Faster!)

Successive Percent Changes ⚠️

Key Takeaways — Part 2

Inverse Variation: $y = k/x$

Joint Variation

SAT Application

Key Takeaways — Part 3

Common Conversions (SAT-relevant)

SAT Unit Conversion Strategy

Key Takeaways — Part 4

Similar Triangles

Map/Model Problems

SAT Application

Key Takeaways — Part 5

Work/Rate Problems

SAT Strategy for Rate Problems

Key Takeaways — Part 6

Common SAT Mistakes

Key Takeaways — Part 7

Ratios, Proportions, and Percents

Ratios, Proportions, and Percents - Complete Interactive Lesson

Part 1: Ratios & Rates

Ratios, Proportions & Percentages

Ratios

Rates

Proportions

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

Percent Change

Multiplier Method (Faster!)

Successive Percent Changes ⚠️

Key Takeaways — Part 2

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

Joint Variation

SAT Application

Key Takeaways — Part 3

Common Conversions (SAT-relevant)

SAT Unit Conversion Strategy

Key Takeaways — Part 4

Similar Triangles

Map/Model Problems

SAT Application

Key Takeaways — Part 5

Work/Rate Problems

SAT Strategy for Rate Problems

Key Takeaways — Part 6

Common SAT Mistakes

Key Takeaways — Part 7

Direct Variation: $y = kx$

Inverse Variation: $y = k/x$