Advanced SAT Problems: Ratios and Percentages

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Tackle complex SAT-style problems that combine ratios, proportions, and percentages. This stack covers combined ratios, population density, margin of error, and other multi-step word problems designed to test deeper understanding.

Advanced SAT Problems: Ratios and Percentages

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The ratio of apples to bananas in a basket is $3:4$, and the ratio of bananas to cherries is $2:5$. What is the ratio of apples to cherries?

Write the ratios as $A:B = 3:4$ and $B:C = 2:5$. To combine them, make the 'B' term equal. Multiply the second ratio by $2$ to get $B:C = 4:10$. Now you can combine them: $A:B:C = 3:4:10$. The ratio of apples to cherries (A:C) is $3:10$.

The ratio of boys to girls in a club is $2:3$. If there are $35$ members in total, how many more girls are there than boys?

The ratio parts sum to $2+3=5$. The total members ($35$) divided by the sum of parts ($5$) gives a multiplier of $7$. Number of boys: $2 \times 7 = 14$. Number of girls: $3 \times 7 = 21$. The difference is $21 - 14 = 7$. There are $7$ more girls than boys.

In a partnership, profits are divided in the ratio $5:3$. The difference between the shares is $\$12,000$. What is the total profit?

The difference in ratio parts is $5-3=2$. This difference of $2$ parts corresponds to $\$12,000$. Therefore, $1$ part is worth $\$12,000 \div 2 = \$6,000$. The total number of parts is $5+3=8$. The total profit is $8 \times \$6,000 = \$48,000$.

A school has a ratio of teachers to students of $1:20$. If $40\%$ of the students are girls, what is the ratio of teachers to girls?

Let $T$ be teachers and $S$ be students. $T:S = 1:20$. Let $G$ be girls. $G = 0.40S$. Substitute $S=20T$ into the second equation: $G = 0.40(20T) = 8T$. The ratio of teachers to girls is $T:G$, which is $T:8T$. Simplified, this is $1:8$.

The price of an item is increased by $25\%$. By what percentage must the new price be decreased to restore the original price?

Let the original price be $\$100$. After a $25\%$ increase, the new price is $\$125$. To get back to $\$100$, the price must be reduced by $\$25$. The percent decrease is calculated from the new price: $\frac{25}{125} \times 100\% = \frac{1}{5} \times 100\% = 20\%$. You must decrease the new price by $20\%$.

What is population density, and how is it expressed as a rate?

Population density is a measurement of population per unit of area. It is a rate expressed as: $$ \text{Population Density} = \frac{\text{Number of People}}{\text{Land Area}} $$ Common units are people per square mile or people per square kilometer.

A city with an area of $50$ square miles has a population of $400,000$ people. What is its population density?

$$ \text{Population Density} = \frac{400,000 \text{ people}}{50 \text{ sq miles}} = 8,000 \text{ people per square mile} $$

A poll of $800$ voters shows a candidate has $54\%$ support with a margin of error of $\pm 3\%$. What is the interval that likely contains the true percentage of support for the candidate?

The margin of error gives a range around the measured value. Lower bound: $54\% - 3\% = 51\%$. Higher bound: $54\% + 3\% = 57\%$. The interval is from $51\%$ to $57\%$.

What is the formula for simple interest, and what does each variable represent?

The formula is $I = P \cdot r \cdot t$, where: - $I$ is the interest earned. - $P$ is the principal amount (the initial sum of money). - $r$ is the annual interest rate (in decimal form). - $t$ is the time the money is invested or borrowed for, in years.

Calculate the simple interest on a loan of $\$2,000$ at an annual interest rate of $5\%$ for $3$ years.

Using $I = P \cdot r \cdot t$: $P = 2000$, $r = 0.05$, $t = 3$. $$ I = 2000 \times 0.05 \times 3 = 100 \times 3 = \$300 $$

How does compound interest differ from simple interest?

Simple interest is calculated only on the principal amount. Compound interest is calculated on the principal amount plus the accumulated interest from previous periods. This leads to exponential growth, often described as 'interest on interest'.

A population of bacteria doubles every hour. If you start with $50$ bacteria, how many will there be after $4$ hours?

This is exponential growth, equivalent to a $100\%$ increase each hour. After 1 hr: $50 \times 2 = 100$ After 2 hrs: $100 \times 2 = 200$ After 3 hrs: $200 \times 2 = 400$ After 4 hrs: $400 \times 2 = 800$ Or using the formula $P(1+r)^t = 50(2)^4 = 50 \times 16 = 800$.

Advanced SAT Problems: Ratios and Percentages

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