Armed Services Vocational Aptitude Battery (ASVAB) Arithmetic Reasoning (AR) Syllabus

What is a proportion and how do you solve one?

A statement that two ratios are equal (a/b = c/d). Solve by cross-multiplying: a x d = b x c, then solve for the unknown.

How do you simplify a ratio such as 12:18?

Divide both terms by their greatest common factor. 12:18 divided by 6 gives 2:3.

How do you split a quantity into a given ratio, e.g., $80 in the ratio 3:5?

Add the parts (3 + 5 = 8), divide the total by that sum (80 / 8 = 10), then multiply: 3 x 10 = $30 and 5 x 10 = $50.

What is the basic rate equation linking distance, rate, and time?

Distance = Rate x Time (d = r x t); so Rate = d/t and Time = d/r.

How do you find average speed for a whole trip?

Divide total distance by total time, NOT by averaging the speeds.

What is a unit rate and how do you find it?

A rate with a denominator of 1 (e.g., miles per hour). Divide the quantity by the number of units (e.g., $6 for 3 lb = $2 per lb).

If two people work together, how do you combine their work rates?

Add their individual rates (jobs per unit time); the time together = 1 divided by the combined rate (e.g., 1/4 + 1/6 = 5/12, so 12/5 hours).

What is the simple interest formula?

I = P x r x t, where P is principal, r is the annual rate (as a decimal), and t is time in years.

What is the compound interest formula for the final amount?

A = P(1 + r/n)^(nt), where n is the number of compounding periods per year, r the annual rate, t the years.

How do you compute the total amount owed or accrued with simple interest?

Total = Principal + Interest = P + (P x r x t) = P(1 + rt).

How do you calculate a discount and the sale price?

Discount = original price x discount rate; Sale price = original price - discount (or original x (1 - rate)).

How do you calculate sales tax and the final total?

Tax = price x tax rate; Total = price + tax (or price x (1 + tax rate)).

How do you calculate percent increase or percent decrease?

Divide the amount of change by the original value, then multiply by 100: (change / original) x 100.

In a word problem, what phrases typically signal addition?

Sum, total, plus, increased by, more than, combined, altogether.

In a word problem, what phrases typically signal subtraction?

Difference, less than, fewer, decreased by, minus, remain, take away.

In a word problem, what phrases signal multiplication and division?

Multiplication: product, times, of, twice, per (for rates). Division: quotient, per, ratio, split equally, out of.

In word problems, what word usually translates to an equals sign?

Is, are, was, were, gives, results in, or yields all indicate equality (=).

How does 'less than' differ in order from normal subtraction wording?

'5 less than x' means x - 5, not 5 - x; the order is reversed compared to how it is read.

What are the general steps to solve a multi-step word problem?

Read carefully, identify what is asked, list known values, translate to an equation, solve step by step, then check the answer against the question.

How do you find the average (arithmetic mean) of a set of numbers?

Add all the values and divide by the count of values.

If you know the average and the number of items, how do you find the total sum?

Multiply the average by the number of items: Sum = Average x Count.

How do you find a missing value when given the desired average?

Multiply the desired average by the total count to get the required sum, then subtract the sum of the known values.

What is a weighted average and when is it used?

An average where values carry different weights; multiply each value by its weight, sum those products, and divide by the total weight (used when groups differ in size).

How do you solve a basic mixture problem (e.g., combining solutions)?

Track the amount of the key ingredient: (amount1 x concentration1) + (amount2 x concentration2) = total amount x final concentration.

What is the median of a data set and how do you find it?

The middle value when numbers are ordered; with an even count, it is the average of the two middle values.

What is the mode of a data set?

The value that appears most frequently; a set can have one mode, multiple modes, or none.

How many inches are in a foot, feet in a yard, and inches in a yard?

12 inches = 1 foot; 3 feet = 1 yard; 36 inches = 1 yard.

What are common U.S. measurement conversions for capacity and weight?

1 gallon = 4 quarts; 1 quart = 2 pints; 1 pint = 2 cups; 1 pound = 16 ounces; 1 ton = 2000 pounds.

How do you convert a quantity between units (e.g., feet to inches)?

Multiply by the conversion factor when going to a smaller unit (more units), and divide when going to a larger unit (fewer units).

What is estimation and a common technique for it?

Finding an approximate answer by rounding numbers to convenient values before computing, used to check reasonableness of results (e.g., 198 x 21 is about 200 x 20 = 4000).