Graduate Record Examinations (GRE) Quantitative Reasoning: Arithmetic and Number Properties Syllabus

How do you compare two fractions $\frac{a}{b}$ and $\frac{c}{d}$ (with positive denominators) without finding a common denominator?

Cross-multiply: $\dfrac{a}{b} > \dfrac{c}{d}$ if and only if $ad > bc$.

For positive fractions less than $1$, how does increasing the numerator versus the denominator affect the value?

Increasing the numerator (denominator fixed) increases the fraction; increasing the denominator (numerator fixed) decreases the fraction.

What happens to a positive proper fraction's value when you add the same positive number to both numerator and denominator?

A proper fraction ($<1$) moves closer to $1$, so it increases. (An improper fraction $>1$ would decrease toward $1$.)

How do you convert a fraction $\frac{a}{b}$ to a percentage?

Compute $\dfrac{a}{b} \times 100\%$. For example, $\dfrac{3}{8} = 0.375 = 37.5\%$.

What is place value, and what does the digit $7$ represent in $4.073$?

Place value is the value a digit holds based on its position. In $4.073$, the $7$ is in the hundredths place, representing $7 \times \frac{1}{100} = 0.07$.

When multiplying decimals, how do you place the decimal point in the product?

Multiply as if they were whole numbers, then place the decimal point so the product has as many decimal places as the total number of decimal places in the two factors combined.

How do you divide by a decimal, e.g. $\frac{6.4}{0.08}$?

Shift the decimal point in both numbers right until the divisor is a whole number: $\dfrac{6.4}{0.08} = \dfrac{640}{8} = 80$.

What is the general formula for percentage change from an old value to a new value?

$\text{percent change} = \dfrac{\text{new} - \text{old}}{\text{old}} \times 100\%$. A positive result is an increase, a negative result a decrease.

How do you express increasing a quantity by $p\%$ as a single multiplier?

Multiply by $\left(1 + \dfrac{p}{100}\right)$. For a $p\%$ decrease, multiply by $\left(1 - \dfrac{p}{100}\right)$.

If a price increases by $20\%$ and then decreases by $20\%$, what is the net percentage change?

Net multiplier $= 1.20 \times 0.80 = 0.96$, a net decrease of $4\%$. Successive percentage changes are not additive.

What single percentage increase is equivalent to two successive increases of $10\%$ and $30\%$?

Multiplier $= 1.10 \times 1.30 = 1.43$, equivalent to a single $43\%$ increase.

After a $25\%$ increase, what percentage decrease returns a quantity to its original value?

Original $\times 1.25$ must be reduced by a factor returning to $1$: decrease $= 1 - \frac{1}{1.25} = 1 - 0.8 = 0.2$, i.e. a $20\%$ decrease.

What is the difference between a ratio and a fraction in setup, e.g. a ratio $a:b$?

A ratio $a:b$ compares two quantities; the actual amounts are $ak$ and $bk$ for some common multiplier $k$. The fraction of the total that is the first part is $\frac{a}{a+b}$.

If a sum of money is divided in the ratio $2:3:5$, what fraction does the largest share represent?

Total parts $= 2+3+5 = 10$, so the largest share is $\dfrac{5}{10} = \dfrac{1}{2}$ of the money.

How do you combine two ratios $a:b$ and $b:c$ into a single ratio $a:b:c$?

Scale them so the shared term $b$ matches in both ratios (using its LCM), then write all three terms together. E.g. $2:3$ and $4:5$ become $8:12:15$.

What is the defining equation of direct proportion between $y$ and $x$?

$y = kx$ for a constant $k$, equivalently $\dfrac{y}{x} = k$. Doubling $x$ doubles $y$; the graph is a line through the origin.

What is the defining equation of inverse proportion between $y$ and $x$?

$y = \dfrac{k}{x}$, equivalently $xy = k$ (constant). Doubling $x$ halves $y$.

If $y$ is inversely proportional to $x$ and $y = 12$ when $x = 4$, find $y$ when $x = 8$.

$k = xy = 4 \times 12 = 48$, so $y = \dfrac{48}{8} = 6$.

What is the fundamental relationship between distance, speed, and time?

$\text{distance} = \text{speed} \times \text{time}$, so $\text{speed} = \dfrac{\text{distance}}{\text{time}}$ and $\text{time} = \dfrac{\text{distance}}{\text{speed}}$.

How do you compute average speed for a whole journey?

$\text{average speed} = \dfrac{\text{total distance}}{\text{total time}}$ — never simply the average of the individual speeds.

A trip covers equal distances at speeds $a$ and $b$. What is the average speed for the whole trip?

The harmonic mean: $\text{average speed} = \dfrac{2ab}{a+b}$.

In work-rate problems, if one worker finishes a job in $a$ hours and another in $b$ hours, how long do they take together?

Combined rate $= \dfrac{1}{a} + \dfrac{1}{b}$, so time together $= \dfrac{ab}{a+b}$ hours.

Two objects move toward each other at speeds $u$ and $v$; what closing (relative) speed do you use, versus moving in the same direction?

Toward each other (or apart): relative speed $= u + v$. Same direction: relative speed $= |u - v|$.

State the product and quotient rules for exponents with the same base.

$a^{m} \cdot a^{n} = a^{m+n}$ and $\dfrac{a^{m}}{a^{n}} = a^{m-n}$.

State the power-of-a-power rule and the power-of-a-product rule for exponents.

$(a^{m})^{n} = a^{mn}$ and $(ab)^{n} = a^{n} b^{n}$.

What do $a^{0}$ and $a^{-n}$ equal (for $a \neq 0$)?

$a^{0} = 1$ and $a^{-n} = \dfrac{1}{a^{n}}$.

How is a fractional exponent $a^{m/n}$ written in radical form?

$a^{m/n} = \sqrt[n]{a^{m}} = \left(\sqrt[n]{a}\right)^{m}$.

What are the product and quotient rules for radicals (with non-negative values)?

$\sqrt{a}\,\sqrt{b} = \sqrt{ab}$ and $\dfrac{\sqrt{a}}{\sqrt{b}} = \sqrt{\dfrac{a}{b}}$.

How do you rationalize the denominator of $\dfrac{1}{\sqrt{5}}$?

Multiply numerator and denominator by $\sqrt{5}$: $\dfrac{1}{\sqrt{5}} = \dfrac{\sqrt{5}}{5}$.

What is the standard form of a number in scientific notation, and write $0.00042$ in it?

Scientific notation is $a \times 10^{n}$ with $1 \leq |a| < 10$ and integer $n$. Thus $0.00042 = 4.2 \times 10^{-4}$.

When multiplying numbers in scientific notation, e.g. $(3 \times 10^{5})(2 \times 10^{-2})$, how do you proceed?

Multiply the coefficients and add the exponents: $3 \times 2 = 6$ and $10^{5} \cdot 10^{-2} = 10^{3}$, giving $6 \times 10^{3}$.