Mathematics Study Guide for the HiSET Test
Page 2
More Numbers and Operations
Ratio and Proportion
A ratio is the quantitative relation between two amounts showing the number of times one value contains or is contained within the other.
Example 1:
A company sold \(24\) electric chainsaws and \(45\) gasoline chainsaws. What is the ratio of electric chainsaws to gasoline chainsaws?
Answer: \(\frac{5}{9}\)
Explanation:
Since the question asks for the ratio of electric chainsaws to gasoline chainsaws, the number of electric chainsaws is in the numerator and the number of gasoline chainsaws is in the denominator. This ratio is simplified as shown below.
\[\frac{{25}}{{45}} = \frac{{5 \cdot 5}}{{5 \cdot 9}} = \frac{5}{9}\]Example 2:
A company has \(24\) male coworkers. There are \(12\) more female coworkers than male. What is the ratio of female coworkers to male coworkers?
Answer: \(\frac{3}{2}\)
Explanation:
Write a ratio with the number of female coworkers in the numerator and the number of male coworkers in the denominator. Then, simplify the ratio, as shown below.
\[\frac{{24 + 12}}{{24}} = \frac{{36}}{{24}} = \frac{{12 \cdot 3}}{{12 \cdot 2}} = \frac{3}{2}\]A proportion is an equation showing two ratios are equal. All proportions are solved by cross multiplying. The rule for cross multiplying is:
\[\begin{array}{c}\frac{a}{b} &=& \frac{c}{d}\\ad &=& bc\end{array}\]When solving a proportion, first, cross multiply. Then divide to isolate the unknown value.
Example:
Solve: \(\frac{3}{7} = \frac{{18}}{x}\)
Answer: \(x=42\)
Explanation:
Begin with the original equation. Cross multiply. Divide by \(3.\)
\[\begin{array}{c}\frac{3}{7} &=& \frac{{18}}{x}\\3x &=& 7 \cdot 18\\3x &=& 126\\x &=& \frac{{126}}{3}\\x &=& 42\end{array}\]Sometimes proportions contain more than one term in the numerator or denominator. When this is the case, cross multiplying involves distribution.
Example:
Solve: \(\frac{{y - 1}}{{y + 2}} = \frac{3}{6}\)
Answer: \(y=4\)
Explanation:
Solve by cross multiplying using distribution. Then move the non-variable terms to the right side of the equation and the variable terms to the left side. Then, isolate the variable.
\[\begin{array}{c}\frac{{y - 1}}{{y + 2}} &=& \frac{3}{6}\\6(y - 1) &=& 3(y + 2)\\6y - 6 &=& 3y + 6\\6y - 3y - 6 + 6 &=& 3y - 3y + 6 + 6\\3y &=& 12\\y &=& 4\end{array}\]Other Number Relationships
We can use mathematics to find other number relationships, such as the total cost of an item after adding sales tax, how much an item would cost if it is on sale, the cost per item, work rate, percents, averages, and rounding. Let’s work on these one at a time.
Working with Money and Discounts
Almost every state has a sales tax. To calculate the amount of the sales tax on an item, use this formula:
\[\text{original}\;\text{price} \cdot \text{sales tax rate} = \;\text{sales tax}\]Example:
How much is the sales tax on a coffee maker that costs \(\$59.99\) if the sales tax rate is \(5.5\% ?\)
Answer: \(\$ 3.30\)
Explanation:
Convert the tax rate from a percent to a decimal: \(5.5\% = 0.055\)
Multiply: \(\$ 59.99 \cdot 0.055 = \$ 3.29945\)
Round the answer to the nearest cent: \(\$3.30\)
The sales tax is \(\$3.30\)
If we are buying an item, and want to know the total price after sales tax is added, we use this formula:
\[\text{total price} = \text{original price} + \text{sales tax}\]Example:
What is the total price of a sleeping bag that costs \(\$79.52\) if the tax rate is \(6.5\%?\)
Answer: \(\$84.69\)
Explanation:
Find the amount of the sales tax: \(\$ 79.52 \cdot 0.065 = \$ 5.1688\)
After rounding the sales tax to the nearest cent, add the original price and the amount of the sales tax: \(\$ 79.52 + \$ 5.17 = \$ 84.69\)
Sometimes, we can find an item on sale. If we see a sign saying the store is taking a certain percent of the price off, we can find the discounted amount using this formula:
\[\text{original price} \cdot \text{discount percent} = \text{amount of discount}\]Then we can find the discounted price using this formula:
\[\text{original price} - \text{amount of discount} = \text{discounted price}\]Example:
A wireless mouse that originally costs \(\$ 31.99\) is discounted \(25\% .\) What is the discounted price?
Answer: \(\$ 23.99\)
Explanation:
Convert the discount percent to a decimal: \(25\% = 0.25\) Use the formula to find the amount of the discount: \(\$ 31.99 \cdot 0.25 = \$ 7.9975\)
After converting the amount of the discount to the nearest cent, subtract the amount of the discount from the original amount: \(\$ 31.99 - \$ 8.00 = \$ 23.99\)
When a store purchases its stocks from a wholesale supply source, the store determines the selling price of each item so the store can make a profit. This process is called marking up the cost. To find the amount of the markup, use this formula:
\[\text{original cost} \cdot \text{markup rate} = \text{amount of markup}\]Then to calculate how much the store will sell the item for, add the original cost and the amount of the markup.
\[\text{original cost} + \text{amount of markup} = \text{selling cost}\]Example:
Suppose a lumber yard buys sheets of plywood for \(\$14.25\) per sheet and sells the sheets after a \(75\%\) markup. What is the selling price of each sheet of plywood?
Answer: \(\$24.94\)
Explanation:
Convert the markup percent to a decimal: \(75\% = 0.75\) Multiply the original cost by the markup decimal to find the amount of the markup: \(\$ 14.25 \cdot 0.75 = \$ 10.6875\) After rounding the amount of the markup to the nearest cent, add the original cost and the amount of the markup: \(\$ 14.25 + \$ 10.69 = \$ 24.94\)
Finding Rate
To a certain degree, our world is filled with rates; how fast we are traveling, the cost per item, how many units of work are being done per hour, mileage, cost per meal, etc. In this section, we will find some of these rates. In most cases, a rate is found by dividing a total by a quantity, or with proportions.
Example 1:
Suppose your car’s gas tank holds \(21\) gallons of gas and you can drive \(399\) miles before running out of gas. What is your car’s mileage rate?
Answer: \(19\) miles per gallon
Explanation:
Calculate mileage rate by dividing the number of miles by the number of gallons: \(399 \div 21 = 19\)
Example 2:
Lisa is buying steak. The package costs \(\$ 23.98\) and weighs \(2.5\) pounds. To the nearest cent, what is the price per pound?
Answer: \(\$ 9.59\)
Explanation:
Calculate the price per pound by dividing the total price by the number of pounds: \(\$ 23.98 \div 2.5 = \$ 9.592\)
Round the answer to the nearest cent.
Example 3:
A package of \(6\) chocolate bars cost \(\$ 11.34\) What is the unit price?
Answer: \(\$1.89\)
Explanation:
To find the unit price, divide the total price by the number of candy bars: \(\$ 11.34 \div 6 = \$ 1.89\)
Example 4:
If Bryan can pick \(18\) flats of strawberries in \(3\) hours, how many flats can he pick in \(8\) hours?
Answer: \(48\)
Explanation:
This is a work rate problem. Set up a proportion: \(\frac{{18}}{3} = \frac{x}{8}\)
Cross multiply and solve the proportion.
Example 5:
If Stacey earns \(\$954\) working \(40\) hours, how much does she earn working \(8\) hours?
Answer: \(\$190.80\)
Explanation:
This is a rate problem. Set up a proportion: \(\frac{{954}}{{40}} = \frac{x}{8}\)
Cross multiply and solve the proportion.
Percent
Percent is considered to be part over whole, which is a ratio. A common formula for finding percent is:
\[\frac{{\text{is}}}{{\text{of}}} = \frac{\% }{{100}}\]We’ll call this the “is over of” formula.
When asked a percent question, simply plug in the given values and solve for the unknown.
Example 1:
What \(\%\) of \(39\) is \(4?\) Round to the nearest tenth of a percent.
Answer: \(10.3\)
Explanation:
Write a proportion using the “is over of” formula above:
\[\frac{4}{{39}} = \frac{\% }{{100}}\]Cross multiply. The answer is \(10.256.\)[.] Round as requested.
Example 2:
\(28\%\) of \(63\) is what number?
Answer: \(17.64\)
Explanation:
Write a proportion using the “is over of” formula above.
\[\frac{x}{{63}} = \frac{28}{{100}}\]Cross multiply. The answer is \(17.64\)[.]
Example 3:
\(3\) is what \(\%\) of \(56.2?\) Round to the nearest tenth of a percent.
Answer: \(5.3\)
Explanation:
Write a proportion using the “is over of” formula above.
\[\frac{3}{{56.2}} = \frac{x}{{100}}\]Cross multiply. Round as requested.
The answer is \(5.3.\)
Example 4:
What is \(13\%\) of \(86?\)
Answer: \(11.18\)
Explanation:
Write a proportion using the “is over of” formula above.
\[\frac{x}{{86}} = \frac{13}{{100}}\]Cross multiply. The answer is \(11.18\)[.]
Another common percent problem is finding the percent of change. The formula for a percent of change operation is:
\[\text{percent change} = \frac{{\text{amount of change}}}{{\text{original amount}}} \times 100\]Example 5:
On Friday evening the number of dads who accompanied their daughters to see an animated show was \(455\). On Saturday evening \(389\) dads brought their daughters to see the same show. What percent decrease is this? Round the answer to the nearest tenth of a percent.
Answer: \(14.5\%\)
Explanation:
Use the formula above:
\[\frac{{455 - 389}}{{455}} = 0.14505495\]Convert to a percent: \(0.14505495 \cdot 100 = 14.5\%\)
Example 5:
Last year, the number of fish in a certain lake was estimated to be \(1320\). This year, the estimate is \(1575.\) What percent increase is this? Round the answer to the nearest tenth of a percent.
Answer: \(19.3\%\)
Explanation:
Use the formula above:
\[\frac{{1575 - 1320}}{{1320}} = 0.193181818\]Convert to a percent: \(0.1931818 \cdot 100 = 19.3\%\)
Average
To find the average of a set of numbers, add the numbers and divide by the number of numbers in the set.
Example:
Find the average of \(89,\,92,\,76,\,82,\,68,\,79,\, \text{and} \,81\)
Answer: \(81\)
Explanation:
The average of the seven numbers is:
\[\frac{{89 + 92 + 76 + 82 + 68 + 79 + 81}}{7} = \frac{{567}}{7} = 81\]Estimation and Rounding
Many times when using mathematics, we need to find an answer quickly, but we don’t need the exact answer, or the exact answer is somewhat cumbersome and confusing. To aid us in this situation we can use estimation or rounding.
First, we will use estimation strategies. In this strategy, we round numbers before we calculate, making the calculations mental math. The actual accuracy of the estimate is not important.
Example 1:
Estimate the sum: \(125 + 586 + 491 + 712\)
Answer: \(1900\)
Explanation:
Round the numbers to one significant digit and add:
\[125 + 586 + 491 + 712 \approx 100 + 600 + 500 + 700 = 1900\](Notice that the exact answer is \(1914.\))
Example 2:
Estimate the product: \(285 \cdot 525\)
Answer: \(150,000\)
Explanation:
Round the numbers to one significant digit and multiply:
\(285 \cdot 525 \approx 300 \cdot 500 = 150,000\) (Notice that the exact answer is \(149,625.\))
Another method of simplifying a numerical answer is rounding. Using this strategy, we combine the numbers first, and then we round the answer.
Example 3:
What is the sum of these numbers to the nearest thousand? \(2541 + 3389 + 6487 + 8692\)
Answer: \(21,000\)
Explanation:
Add the numbers: \(2541 + 3389 + 6487 + 8692 = 21,109.\) Round the answer to the nearest thousand.
Example 4:
What is the product of these numbers to the nearest ten-thousand? \(2541 \cdot 3389\)
Answer: \(8,610,000\)
Explanation:
Multiply the numbers: \(2541 \cdot 3389 = 8,611,449.\) Round the answer to the nearest ten-thousand.
Multi-step Problems
Solve these problems. Each problem takes more than one step.
Example 1:
Jessica discovered a bag of precious stones. She opened it up and found that it contained \(148\) diamonds and \(213\) rubies. She sold \(67\) gems and bought \(46\) different ones. How many gems does Jessica have now?
Answer: \(340\)
Explanation:
Add the number of gems in the bag, subtract the quantity she sold, and then add the quantity she bought. \(148 + 213 - 67 + 46 = 340\)
Example 2:
Ahmed has \(200\) coins. Half of them are quarters and one-fourth of the remaining coins are dimes. One-third of the rest of the coins are nickels, and the rest of the coins are pennies. How many pennies does Ahmed have?
Answer: \(50\)
Explanation:
Ahmed starts with \(200\) coins. If one-half of the coins are quarters, then that leaves \(100\) other coins. Then, if one-fourth of those coins are dimes, that means he has \(25\) dimes. This leaves \(75\) coins. Now, if one-third of these coins are nickels, that means Ahmed has \(25\) nickels. That leaves \(50\) pennies.
Example 3:
Harvey bought \(4\) packs of red plastic balls, \(3\) packs of green plastic balls, \(2\) packs of blue plastic balls, and \(5\) packs of yellow plastic balls. Each pack of plastic balls has \(24\) balls in them. One-eighth of the balls were defective. How many balls were defective?
Answer: \(42\)
Explanation:
Calculate the total number of balls Harvey bought.
\(4(24) + 3(24) + 2(24) + 5(24) = 336\) Divide \(336\) by \(8\). The result is \(42.\)
Example 4:
One man can paint a house by himself in \(18\) hours. Another man can paint the house by himself in \(36\) hours. How long will it take for them to paint the house working together?
Answer: \(12\) hours
Explanation:
This is a rational equation work problem. Write the following equation, which shows the two men’s times, with \(x\) being their time together.
\[\frac{1}{{18}} + \frac{1}{{36}} = \frac{1}{x}\]Multiply by the LCD, which is \(36x.\)
\[36x\left( {\frac{1}{{18}} + \frac{1}{{36}} = \frac{1}{x}} \right)\]Simplify.
\[2x + x = 36\]Combine like terms and solve for \(x.\) The answer is \(12\) hours.
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