Cracking SAT Math: Percentages & Proportions Explained

Welcome to understanding percentages! Today we'll explore how percentages help us work with parts of a whole.A percentage literally means 'per hundred'. Think of it as dividing something into one hundred equal parts.For example, twenty-five percent means twenty-five out of these hundred squares.Let's look at how percentages relate to decimals and fractions.To convert a decimal to a percentage, we move the decimal point two places right and add the percent symbol.Some percentages are especially useful as benchmarks for mental math.Let's apply this to a real-world example. If an eighty dollar item is twenty-five percent off, we can quickly calculate the discount using our benchmark knowledge.Here are some quick mental math tips for calculating common percentages.To calculate percentage change, we use this formula: the change in value divided by the original value, times one hundred.Let's look at a price increase example. A product's price increased from eighty dollars to one hundred dollars.To find the percentage increase, we first calculate the change: twenty dollars. Then divide by the original price and multiply by one hundred.Now let's examine population growth. A city's population grew from fifty thousand to fifty-seven thousand five hundred.The change is seven thousand five hundred. Dividing by the original population and multiplying by one hundred gives us fifteen percent growth.A common mistake is adding or subtracting percentages directly. Let's see why this is incorrect.The correct approach is to multiply by percentage factors. For example, a twenty percent increase followed by a thirty percent increase.Let's solve an SAT-style problem. A shirt's price changed multiple times, ending at eighty-four dollars.To find the original price, we first calculate the total percentage factor, then divide the final price by this factor.Finally, let's look at how to calculate multiple discounts. A store offers forty percent off, then an additional twenty-five percent off.We multiply the remaining percentages: sixty percent times seventy-five percent equals forty-five percent of the original price, meaning a total discount of fifty-five percent.A proportion is an equation showing that two ratios are equal.Cross multiplication is a powerful technique where we multiply the numerator of one fraction by the denominator of the other.Let's solve a percentage problem using proportions. We want to find what percent 15 is of 60.Cross multiply to get fifteen times one hundred equals sixty x.Simplify the left side.Solve for x to find that 15 is 25 percent of 60.Let's solve a map scale problem using proportions.We set up the proportion comparing the scale to our actual measurements.Cross multiply and solve to find that the cities are 125 miles apart.Similar figures have the same shape but different sizes. Their corresponding sides form proportions.The ratios of corresponding sides are equal. We can use this to find missing measurements.Let's solve a mixture problem involving different concentrations of salt solution.Let x represent the amount of twenty percent solution we need to add.The amount of salt in the parts equals the amount of salt in the whole mixture.Simplify and solve for x.Let's solve a complex problem involving both percentage changes and proportions.First multiply by one point two for the increase, then by point seven five for the discount.Simplify to find that the original price was one hundred dollars.Let's review the key points about working with proportions and cross multiplication.Remember these important strategies for solving proportion problems effectively.Keep practicing these techniques to become confident with proportions and cross multiplication!