#AP SAT (Digital) Math: Percentages, Fractions, and Decimals - Your Ultimate Guide 🚀

Hey there, future SAT superstar! This guide is your go-to resource for acing those percentage, fraction, and decimal questions. Let's get you feeling confident and ready to rock the exam!

#1. The Power of Percentages

Percentages are all about parts of a whole, and they're super common on the SAT. Think of them as a way to express fractions with a denominator of 100. Let's break it down:

#1.1. Understanding the Basics

What is a Percentage? It's a number out of 100, shown with the "%" symbol. So, 50% means 50 out of 100. * Key Idea: The "whole" is always 100%, and the "part" is some percentage of that whole.

Key Concept

Finding a Percentage of a Quantity:

Convert the percentage to a decimal (divide by 100).
Multiply the decimal by the quantity.

Example: What is 25% of 80? 25% = 0.25. 0.25 * 80 = 20.

Memory Aid

The "Is over Of" Trick:

Is/Of = %/100
If you're asked, "What percent of 50 is 10?" think: 10/50 = x/100. Solve for x!

#1.2. Practical Applications

Discounts: 30% off a $100 item means a$ 30 discount ( $100 * 0.30 =$ 30).
Sales Tax: 7% tax on a $50 purchase is$ 3.50 ( $50 * 0.07 =$ 3.50).
Interest: 4% interest on $2000 savings is$ 80 ( $2000 * 0.04 =$ 80).
Data Analysis: If 60 out of 200 students like pizza, that's 30% (60/200 * 100 = 30%).

Exam Tip

Quick Tip: To find 10% of a number, just move the decimal one place to the left. Then, you can easily find 5%, 20%, etc.

#2. Percent Increase and Decrease

Understanding how to calculate changes is crucial for many SAT problems. Let's dive in!

#2.1. Calculating Percent Changes

Percent Increase: ((New Value - Original Value) / Original Value) * 100
Percent Decrease: ((Original Value - New Value) / Original Value) * 100

Key Concept

Finding the New Value Directly:

After Increase: Original Value * (1 + Percentage as a decimal)
After Decrease: Original Value * (1 - Percentage as a decimal)

Example: A $50 item increases by 20%. New price:$ 50 * (1 + 0.20) = $60. </key_point>$

#2.2. Combining Successive Percent Changes

This one can be tricky, but we've got you covered:

Convert each percentage to a decimal.
Add 1 to each of those decimals.
Multiply the results together.
Subtract 1 from the product.
Convert back to a percentage.

<memory_aid> Avoid the Trap: You can't just add or subtract percentages directly when they're applied successively. Always use the method above! </memory_aid>

#2.3. Real-World Applications

Population Growth: A town grows by 10% in one year and then 5% the next. Use the method above to find the total growth.
Stock Market: A stock drops by 15% one day and then increases by 10% the next. What's the net change?
Sales Performance: A company's sales increase by 20% one quarter and then decrease by 5% the next. What's the overall change?

<common_mistake> Common Mistake: Forgetting to use the original value when calculating percent change. Always compare to the starting point! </common_mistake>

#3. <high_value_topic> Fractions, Decimals, and Percentages: The Trio </high_value_topic>

These three are like the best of friends, and you need to be able to switch between them easily.

#3.1. Converting Between Representations

Fraction to Decimal: Divide the numerator by the denominator. (e.g., 3/4 = 0.75)
Decimal to Percentage: Multiply by 100 and add the "%" symbol. (e.g., 0.75 = 75%)
Percentage to Decimal: Divide by 100 (or move the decimal point two places to the left). (e.g., 75% = 0.75)
Decimal to Fraction: Write the decimal as a fraction over 1, then multiply both by 10 for each decimal place. (e.g., 0.25 = 25/100 = 1/4)
Fraction to Percentage: First, convert the fraction to a decimal, then multiply by 100 and add the "%" symbol.

<quick_fact> Quick Fact: Memorize common conversions like 1/2 = 0.5 = 50%, 1/4 = 0.25 = 25%, and 1/5 = 0.2 = 20%. It'll save you time! </quick_fact>

#3.2. Practical Uses and Examples

Cooking: Use fractions in recipes (e.g., 1/3 cup of flour).
Finance: Decimals are everywhere in money (e.g.,12.50).
Grades: Percentages are used for scores (e.g., 88% on a test).
Comparisons: 0.6 = 3/5 = 60% are all the same value.
Probability: A 50% chance of rain is the same as 0.5 or 1/2.
Exam Tip

Exam Tip: When you see a problem with fractions, decimals, and percentages, convert everything to the same format to make it easier to solve. Usually, decimals are easiest for calculations.

#Final Exam Focus 🎯

Okay, let's get down to the nitty-gritty. Here's what to focus on the most:

High-Value Topics:
- Calculating percentages of quantities
- Percent increase and decrease
- Converting between fractions, decimals, and percentages
Common Question Types:
- Word problems involving discounts, taxes, and interest
- Problems with successive percent changes
- Questions that require you to convert between representations
Time Management: Don't spend too long on any one question. If you're stuck, move on and come back later.
Common Pitfalls:
- Forgetting to use the original value in percent change calculations
- Incorrectly converting between fractions, decimals, and percentages
- Adding or subtracting percentages directly when they're applied successively

Exam Tip

Last-Minute Tip: Take a deep breath, review your notes, and remember that you've got this! Confidence is key.

#Practice Questions

Let's put your knowledge to the test! Here are some practice questions to get you warmed up:

Practice Question

#Multiple Choice Questions

A store is having a 20% off sale. If a shirt originally costs $40, what is the sale price? (A)$ 8 (B) $32 (C)$ 48 (D) $50$
A population of bacteria increases by 10% each hour. If there are initially 100 bacteria, how many will there be after 2 hours? (A) 110 (B) 120 (C) 121 (D) 125
What is 0.65 as a fraction in simplest form? (A) 13/20 (B) 65/100 (C) 13/25 (D) 6/10

#Free Response Question

A company's sales increased by 15% in the first quarter and then decreased by 10% in the second quarter. If the sales in the first quarter were200,000:

a) What were the sales at the end of the first quarter? (2 points)
b) What were the sales at the end of the second quarter? (2 points)
c) What was the overall percentage change in sales from the beginning of the first quarter to the end of the second quarter? (3 points)

Scoring Breakdown:

(a) $200,000 * 1.15 =$ 230,000 (2 points: 1 for correct setup, 1 for correct answer)
(b) $230,000 * 0.90 =$ 207,000 (2 points: 1 for correct setup, 1 for correct answer)
(c)
- Calculate the overall factor: 1.15 * 0.90 = 1.035 (1 point)
- Calculate the overall change: 1.035 - 1 = 0.035 (1 point)
- Convert to a percentage: 0.035 * 100 = 3.5% increase (1 point)