Fractions, Decimals, & Percents

Simplifying fractions

multiplying or dividing both the number and denominator by the same number does not change the value of the fraction

Numerator and Denominator Rules

only apply to positive fractions 1. As numerator goes up, the fraction increases 2. As denominator goes up, the fraction decreases 3. Adding same number to both brings fraction closer to 1 Proper fraction increases in value as it approaches 1 Improper fraction decreases in value as it approaches 1

The Unknown Multiplier

technique to solve ratio problems that asks to extract part of a ratio to the full quantity. Use when neither quantity in the ratio is already equal to a number or a variable expression.

Data Sufficiency Answer Choices

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient. (E) Statements (1) and (2) TOGETHER are NOT sufficient.

Methodical Data Sufficiency Approach

*Step 1:* Separate additional info from the actual question *Step 2:* Determine whether the question is Value or Yes/No *Step 3:* Decide exactly what the question is asking (One value...definitive yes OR no) *Step 4:*Use the Grid to evaluate the statements (AD BCE)

1/100

0.01, 1%

1/50

0.02, 2%

1/25

0.04, 4%

1/20

0.05, 5%

1/10

0.10, 10%

1/9

0.111, 11.1%

1/8

0.125. 12.5%

1/6

0.167, 16.7&

1/5

0.20, 20%

1/4

0.25, 25%

3/10

0.30, 30%

1/3

0.333, 33.3%

3/8

0.375, 37.5%

2/5

0.40, 40%

1/2

0.50, 50%

3/5

0.60, 60%

5/8

0.625, 62.5%

2/3

0.666, 66.6%

7/10

0.70, 70%

3/4

0.75, 75%

4/5

0.80, 80%

5/6

0.833, 83.3%

7/8

0.875, 87.5%

9/10

0.90, 90%

Dividing fractions

1. change the divisor into its reciprocal, and then 2. multiply the fractions Note: The divisor is the second number The product of a number and it's reciprocal always equal 1. NEVER split the denominator with terms that incorporate sums or differences

Adding or Subtracting fractions

1. find a common denominator 2. change each fraction so that it is expressing using this common denominator 3. add up the numerators only

1/1

1.00, 100%

5/4

1.25, 125%

4/3

1.333, 133.33%

3/2

1.50, 150%

7/4

1.75, 175%

Benchmark Values: Fractions

1/10, 1/5, 1/4, 1/3, 1/2, 2/3, and 3/4 Rounding errors should cancel each other by rounding some numbers up and others down.

Percent Change vs. Percent of Original

A percent change can always be rephrased as a percent of the original, and vice versa : "increase" equivalent to "greater than" "decrease" equivalent to "less than"

Multiplying fractions

Always to try to cancel factors first

When is it best to use fractions?

Canceling factors in multiplication; expressing proportions that do not have clean decimal equivalents (such as 1/7)

Successive Percents

Cannot simply be added together. Instead, lump the percents into one calculation, rewriting the percent changes as percents of original

Multiple Ratios (2 ratios containing a common element)

Change ratios to have common terms corresponding to the same quantity.

Estimating Decimal Equivalents of Fractions

Choice 1: Make the denomiator the nearest factor or another power of 10 Choice 2: Change the numerator or denominator to make the fraction simply easily. *try to not change both, especially in opposite direction. Small percent adjustments: 100,000 / 96...changing 96 to 100 (4% increase) so to be more exact, calculate: 100,000 / 100 = 1,000 * 1.04 = 1,040

Interest Formula

Compound Interest = P(1 + r/n)^nt P= principal r = rate (in decimal form) n = number of times per year t = number of years Shortcut -> solve with successive percents

Comparing Fractions

Cross-Multiply and put each answer by the corresponding numerator.

Dividing Decimals

Divide by whole numbers. Shortcut -> move decimals in the same direction

Convert Fraction to Decimal

Divide the numerator by the denominator: 3 / 8 = 0.375 Use long division if necessary

When Not to use Smart Numbers

Do not pick smart numbers when any amount or total is given...only when no amounts are given in the problem.

Ratios

Express a particular relationship between two or more quantities by: 1. Using the word "to", as in 3 to 4 2. Using a colon, as in 3:4 3. By writing a fraction, as in 3/4 (only for ratios of 2 quantities)

Last Digit Shortcut

Find a units digit, or a remainder after division by 10? Only pay attention to the units digit of the numbers you're working with.

Concrete Values

If a DS question asks for the concrete value of one element of a ratio, you will need BOTH the concrete value of another element of the ratio AND the relative value of the two elements of the ratio.

Relative Values

If a DS question asks for the relative value of two pieces of a ratio, ANY statement that gives the relative value of ANY two pieces of that ratio will be sufficient.

Terminating Decimals

If, after being full reduced, the denominator has any prime factors besides 2 or 5, the decimal will not terminate The denominator can only have factors of 2 and/or 5 for the decimal to terminate.

Multiplying Decimals

In the factors, count all the digits to the right of the decimal point - then put that many digits to the right of the decimal point in the product. Shortcut -> move decimals in the opposite direction

Adding & Subtracting Decimals

Line up the decimal points

Repeating Decimals

Long division can determine repeating cycle. Common patterns: 4/9 = 0.4444444 23/99 = 0.232323 1/11 = 9/99 = 0.090909 3/11 = 27/99 = 0.272727 If denominator is 9, 99, 999 or another number equall to a power of 10 minus 1, then the numerator gives the repeating digits (perhaps with leading zeroes)

Percents as Decimals

Move the decimal point 2 spaces to the left. Remember the % is always bigger than the decimal.

Trading decimal places for powers of 10

Multiply by positive power of 10 -> move decimal to the right Divide by positive power of 10 -> move decimal to the left. Negative powers of 10 reverse the process

Heavy Division Shortcut (involving decimals)

Only need approximate solution? get a single digit to the left of the decimal in the denominator (remember to shift numerator also) and then focus only on the whole numbers

Percent Increase and Decrease

Original + Change = New Change in percent or change in actual value: Percent Change = Change in Value / Original Value New percentage or new value: New Percent = New Value / Original Value

Percent, Of, Is, What

Percent = divide by 100 (/100) Of = multiply (x) Is = equals (=) What = unknown value (x)

Unknown Digit Problems ("brainteasers")

Principles: 1) Look at the answer choices first, to limit your search. 2) Use other given constraints to rule out additional possibilities. 3) Focus on the units digit in the product or sum. This unit digit is affected by the fewest other digits 4) Test the remaining answer choices

Mixture Chart

Problems calling for Chemical Mixtures and Percents & Weighted Averages. Volume (mL) Original Change New -------------------------------------------------- Liquid 1 Liquid 2 Total

Powers and Roots

Rewrite the decimal as the product of an integer and a power of 10, and then apply the exponent: (0.5)^4= 0.5 = 5 * 10^-1 (5 * 10^-1)^4 = 5^4 * 10^-4 5^4 = 25^2 = 625 625 * 10^-4 = 0.0625 Shortcut -> count decimal places in original decimal, then: for EXPONENTS, multiply that by the exponent to get final number of decimal places for ROOTS divide that by the root to get the final number of decimal places.

True/False: Ratios can express a part-part relationship or a part-whole relationship

True. part-part relationship: the ratio of men to women in the office is 3:4. part-whole relationship: there are 3 men for every 7 employees If there are only two parts in the whole, you can derive a part-whole ratio from a part-part ratio, and vice versa.

Using Place Value

Unknown digit problems - create variables (such as x, y, and z) to represent the unknown digits. Recognize that each unknown is restricted to at most 10 possible values (0-9). Then apply any given constraints, which may involve number properties such as divisibility or odds & evens.

Smart Numbers for Percents

Unspecified numerical amounts (possibly often described by variables)? Pick 100 for the unspecified amounts

Smart Numbers for Fractions

Unspecified numerical amounts (possibly often described by variables)? Pick common multiples of the denominators

General Rule for using fractions or decimals/percents

Use fractions for multiplication or division Prefer decimals and percents for addition/subtraction, estimating numbers, or comparing numbers.

Convert Decimal to Fraction

Use place value of the last digit in the decimal as the denominator, and put the decimal's digits in the numerator. Then simply: 0.375 -> 375/1000 = 3/8

Proportions

Used to solve simple ratio problems. 1) Set up a labeled proportion 2) Cross-multiply to solve Shortcut -> cancel factors before cross-multiplying. *Can cancel factors either vertically within a fraction or horizontally across an equals sign. Never cancel diagonally across an equals sign.*

When is it best to use decimals/percents?

addition/subtraction; estimating results; comparing sizes; (b/c basis of comparison is equivalent since there is no denominator).

Fraction

expresses a part-whole relationship in terms of a numerator (the part) and a denominator (the whole).

Decimal

expresses a part-whole relationship in terms of place value (a tenth, a hundredth, a thousandth, etc.)

Percent

expresses a special part-whole relationship between a number (the part) and one hundred (the whole)

Proper fractions

fall between 0 and 1

Improper fractions

greater than 1 can be rewritten as mixed numbers