Math: Numbers, Ratios and Percents

This is Unit Seven, part two, of ACT Preparation. Part one closed by comparing unit rates; this part builds on ratios to reach proportions and percents. The discipline is unchanged: follow the rule for the form in front of you. Here the rules are short, and the errors hide in which number is the base and whether the units match. By the end you will be able to solve a proportion with matching units. To convert among percents, decimals and fractions. To use the percent equation, including finding a whole from a part. To find a percent change and apply tax and discount in order. And to combine successive changes by multiplying.

Two entries on proportions, and on the unit conversion that has to come first.

Two equal ratios.

ACT 0622 Proportions

A proportion states that two ratios are equal, and it is solved by cross-multiplying: if a over b equals c over d, then a d equals b c. X over twelve equals five over eight gives 8x equals sixty, so x is 7.5. Put matching units in matching positions: miles over hours on both sides.

A proportion word problem, step by step.

ACT 0623 Solving a Proportion With Units

Identify the two quantities in a fixed ratio, convert each to one unit, write the proportion with the same quantity on top on both sides, then cross-multiply and check the answer's units. A recipe uses three cups of flour for twenty-four cookies; five dozen is sixty cookies, so three over twenty-four equals x over sixty, and x is 7.5 cups. On a map where two inches represent fifteen miles, a road of 52.5 miles is seven inches. The conversion is where the error hides: leaving five dozen as five gives 0.625 cups.

Four entries on percent and the percent equation.

What percent means.

ACT 0624 Percent Means per Hundred

Percent means per hundred. To change a percent to a decimal, divide by one hundred; to a fraction, write it over one hundred and simplify. Thirty-five percent is 0.35, or seven twentieths, and a percent can exceed one hundred: 150 percent is 1.5. Small percents trip the conversion: 0.4 percent is 0.004.

Comparing numbers in different forms, step by step.

ACT 0625 Comparing Numbers Written Different Ways

Choose one form, usually the decimal, convert every number to it, then line the decimals up by place value and compare from the left. Three eighths is 0.375, thirty-seven percent is 0.37, and 3.5 times ten to the negative one is 0.35; with 0.36, the order from least is 0.35, 0.36, 0.37, 0.375. Write enough decimal places: rounded to one place, all four look like 0.4, and the order disappears.

Amount, percent and base.

ACT 0626 The Percent Equation

The percent equation is amount equals percent times base, with the percent written as a decimal, and any one of the three can be found from the other two. Fifteen percent of eighty is twelve. Eighteen is what percent of seventy-two? Eighteen divided by seventy-two is 0.25, or twenty-five percent. The base is the number the percent is taken of.

Finding the whole from a part, step by step.

ACT 0627 Finding the Whole From a Part

Identify the amount and the percent it represents, write the equation with the base unknown, divide the amount by the decimal percent, and check. Forty-two students who walk are thirty-five percent of the student body: forty-two divided by 0.35 is 120, and thirty-five percent of 120 is forty-two. A sale price of $68 at eighty-five percent of the original means an original of $80. Multiplying instead gives 14.7, smaller than the part, so it cannot be the whole.

Four entries on percent change, and on tax, discount and interest.

Dividing by the original.

ACT 0628 Percent Change: Divide by the Original

Percent change is the amount of change divided by the original amount. A price rising from $40 to $50 changes by $10, and ten divided by forty is twenty-five percent. Divide by the original, never the new amount: ten divided by fifty, twenty percent, answers a different question, the decrease from fifty back to forty.

Percent change, step by step.

ACT 0629 Percent Change, Step by Step

Identify the original, the earlier value, and the new amount; subtract to find the change; divide by the original and name the direction. A town falling from 2,500 to 2,150 people loses 350, and 350 divided by 2,500 is a fourteen percent decrease. To find a new amount, multiply by one plus the rate or one minus the rate: a $60 jacket marked up fifteen percent costs sixty times 1.15, $69. The original is the earlier value, whichever is larger.

Tax, discount, mark-up, commission and interest.

ACT 0630 Tax, Discount, Mark-up, Commission and Interest

Sales tax, discount, mark-up and commission are each a percent of a base: the price, the original price, the cost, or the sales. Simple interest is principal times rate times time in years: $2,000 at four percent for three years earns $240. Time must be in years, so six months is 0.5.

A discount, then a tax, step by step.

ACT 0631 A Discount Then a Tax

Find the sale price first, then apply the tax to the sale price, multiplying by one plus the tax rate, and round only at the end. An $80 pair of shoes at twenty-five percent off is $60, and six percent tax makes $63.60. Tax applies to the price actually paid: six percent of $60 is $3.60, while six percent of the $80 original would be $4.80, a tax on money never spent.

Three entries on changes in a row, and the whole unit in one trip.

Why two changes do not add.

ACT 0632 Successive Percent Changes Do Not Add

Two percent changes in a row multiply; they do not add. Up twenty percent then down twenty percent is 1.2 times 0.8, which is 0.96, a four percent loss, not zero change. The second percent is taken of a new base, so the changes cannot be added.

Combining two changes, step by step.

ACT 0633 Two Changes in a Row

Write each change as a multiplier, one plus r or one minus r, multiply the multipliers, and read the overall change from the product. A stock up twenty-five percent and then down twenty percent: 1.25 times 0.8 is exactly one, no change, though twenty-five minus twenty suggests a five percent gain. Ten percent twice is 1.21, a twenty-one percent increase. To undo a change, divide by its multiplier: after a twenty percent drop, recovery needs twenty-five percent, because one divided by 0.8 is 1.25.

The whole unit in one trip.

ACT 0634 One Trip, Three Tools

This unit's tools share one discipline: follow the rule for the form in front of you. Numbers nest; the order of operations and the exponent rules govern every computation; radicals simplify by factoring out perfect squares; i squared is negative one. Ratios compare by division, proportions cross-multiply with matching units, percent means per hundred, percent change divides by the original, and successive changes multiply. A family drives 312 miles in six hours on twelve gallons: fifty-two miles per hour, twenty-six miles per gallon. The return is 273 miles, so a proportion gives 10.5 gallons. Gas at $3.40 with a five percent discount is $3.23 a gallon, and the return's fuel costs about $33.92. The return used 1.5 gallons less, a decrease of 1.5 divided by twelve, 12.5 percent. Convert units first, divide by the original, multiply successive changes, and round only at the end.

Look at what you can now do. You can solve a proportion by cross-multiplying, converting units before you set it up. You can move among percents, decimals and fractions, and order numbers by writing enough decimal places. You can use amount equals percent times base, and divide to find a whole. You can find a percent change by dividing by the original, and tax the price actually paid. And you can combine changes by multiplying their multipliers, and undo one by dividing. Unit eight turns from numbers to expressions: polynomials and factoring.