Week 7, Day 1 and 2: Ratios and Math Formulas Review

Chapter 11: Math Foundations – Ratios and Math Formulas Review

 

Ratios

Fractions

Treating ratios as fractions is the easiest way to make comparisons. Imagine a picture comparing parts to the whole.

 

Equivalent Fractions

Multiply the numerator and the denominator by the same number (1/1 = 2/2 = 3/3 etc.) to create the equivalent fraction. Divide by the same number to reduce the fraction to lowest terms.

 

Reducing (Canceling)

Since most fractions on the GRE are given in lowest terms, it is useful to cancel where possible.


Adding and Subtracting Fractions

Fractions must have the same denominator to add or subtract the numerators. The least common denominator is the best choice.

 

Multiplying Fractions

Multiplying fractions requires you to multiply numerator to numerator and denominator to denominator.

 

Reciprocals

Invert the fraction – turn it upside-down – to get the reciprocal. The reciprocal of a whole number is a fraction with that number as the denominator and 1 as the numerator.

 

Dividing Common Fractions

Multiply by reciprocals to divide fractions!

 

Comparing Positive Fractions

When fractions have the same denominator, choose the one with the larger numerator as the larger fraction; when they have different denominators but the same numerator, the smaller denominator has the larger value. If neither is the same, you can do one of three things: change both to decimal equivalents; find the least common denominator; or cross multiply.

 

Mixed Numbers and Improper Fractions

Terminology: mixed numbers include an integer and a fraction, but improper fractions are fractions whose numerators are larger than their denominators. We have to be able to convert these back and forth. Divide improper fractions to create mixed numbers; use multiplication to change mixed numbers into fractions.

 

Properties of Fractions Between -1 and +1

Reciprocals of any fraction between 0 and 1 create numbers larger than both the original fraction and 1. Reciprocals of fractions between -1 and 0 create numbers smaller than both the original fractions and -1. Squares of fractions between 0 and 1 are smaller, while squares of fractions between 0 and -1 are larger. Multiplying any positive number by a fraction between 0 and 1 creates a smaller product, but multiplying any negative number by a fraction between 0 and 1 gives a product greater than the original.

 

Decimals

Converting Decimals

Convert a fraction to a decimal by doing division; convert a decimal to a fraction by treating it as a number over the place value.

 

Comparing Decimals

Line up the decimal point for comparisons; larger numbers have a larger value closest to the decimal point.

 

Estimation and Rounding

“Round up” a decimal that starts with 5, 6, 7, 8, or 9; “round down” a decimal that starts with 0, 1, 2, 3, or 4. Estimations are useful throughout the GRE – just be sure to check the answers to see how close your estimation needs to be.

 

 

Percents

A percent is a fraction out of a hundred; solve the problems by treating each fraction or decimal equivalent as a part over a whole times 100%.

 

Translating English to Math in Part-Whole Problems

Avoid careless errors by carefully looking at “is” and “of” – “is/are” introduces the part, and “of” introduces the whole.

 

Properties of 100%

100% = 100/100 – all the parts of the whole should add up to equal 100. Multiplying or dividing by 100% is the same thing as multiplying or dividing by 1.

 

Converting Percents

To change a fraction to its percent equivalent, multiply by 100%; to change a decimal to its percent equivalent, move the decimal to the right two places. To change a percent to a fraction, divide by 100%; to change a percent to a decimal, move the decimal to the left two places. When you divide a percent by a percent, you drop the percent sign, but when you divide a percent by a number, the percent sign stays.

 

Common Percent Equivalents

Memorize these! So useful! (p. 216 in 2020 book)

 

Using the Percent Formula to Solve Percent Problems

Part/Whole (100%) = percent.

 

Percent Increase and Decrease

Percent change = change/starting point, and change is always the difference between the original and the new value.

 

Multistep Percent Problems

Remember that the percent of a percentage is NOT the same as a percent of the original base number. You must do these problems step-by-step.

 

Picking Numbers with Percents

Pick 100 as equal to 100% when the whole is unknown. This makes calculations easier.

 

Percent Word Problems

Identify percent, whole, and part using keywords; profit equals the seller’s price minus the cost to the seller, discount equals original price minus reduced price, and the sale price equals the final price after discount or decrease.


Simple and Compound Interest

Interest is calculation on the basis of a rate of increase on some principal, usually given as a decimal value which can be treated as a percentage. Interest = Principal * Rate * Time or I = PRT. Principal is the amount being invested. The final value of the account is principal plus interest earned.

Compound interest is a repetition of the process given above, where you compound (add in the interest to the principal) at the end of each time period.

 

 

Ratios

Part:Whole Ratios

Part =  subset; whole = set. Fractions are part to whole ratios.

 

Ratio vs. Actual Number

Ratios are usually reduced to their simplest form. You must know the total whole and the ratio between subsets to calculate the value of the subsets.

 

Ratios of More than Two Terms

The key is to set the ratios equal to each other separately and then to cross multiply and simplify with the desired ratio on one side of the equal sign.

 

 

Math Formulas

Rates

Speed

Speed, velocity, rate is equal to distance divided by time; distance equals rate times time; time equals distance divided by rate.

Other Rates

Time = quantity divided by rate.

 

Combined Rate Problems

Rates can be added to each other to get to a combined rate.

 

 

Combined Work Formula

1/A + 1/B = 1/Total work rate ß take the reciprocal to get the total work rate.

T = (a*b)/(a + b)

T = (a*b*c)/(ab + bc + ac)


Averages

Using the Average to Find a Missing Number

Average = sum / number of terms

Avg = (sum + unknown) / number of terms

 

Another Way to Find a Missing Number: The Concept of “Balanced Value”

The sum of the differences between each term and the mean must equal zero. So you can subtract the mean from each term including x, and set the whole sum equal to zero. Then solve for x.

 

Average of Consecutive, Evenly Spaced Numbers

The average will be the middle value (median) when the numbers are consecutive and evenly spaced. Be careful: NOT all lists of consecutive numbers are evenly spaced (e.g., primes, squares, etc.)

 

Combining Averages

ONLY combine averages when both sets have an even number of terms. Other averages cannot be combined this way, and you have to do weighted averages.


 


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