2.3 Word & quantitative reasoning problems

Setting up multi-step word problems

Quantitative items on the DoD Cyber Test are rarely one calculation; they chain two or three. The winning habit is to write down what you know, note the target, and track units at every step so the answer lands in the right unit.

A four-step routine

1. List the givens with their units.
2. Identify the target quantity and its unit.
3. Choose a bridge, a rate, ratio, or conversion factor that connects givens to target.
4. Cancel units as you multiply so only the target unit survives.

Rates and throughput

A rate is a quantity per unit of something else, such as files per minute or megabits per second. Throughput and bandwidth problems are rate problems in disguise.

Worked example: A link transfers a nine hundred megabyte file in two minutes. What is the throughput in megabytes per second? Two minutes is one hundred twenty seconds. Nine hundred megabytes divided by one hundred twenty seconds equals seven and one-half megabytes per second. If a second file is one thousand five hundred megabytes, dividing by that same seven and one-half megabytes per second gives two hundred seconds, or three minutes and twenty seconds.

The bits-versus-bytes trap: Network speeds are quoted in bits per second, but file sizes are in bytes, and one byte is eight bits. A one hundred megabit-per-second link does not move one hundred megabytes per second; divide by eight to get twelve and one-half megabytes per second. Forgetting this factor of eight is the most common quantitative error on cyber tests.

Ratios and proportions

A ratio compares two quantities; a proportion sets two ratios equal so you can solve for a missing term.

Worked example: A server handles three requests every four milliseconds. How long for one hundred twenty requests? Set up the proportion: three requests over four milliseconds equals one hundred twenty requests over x. Cross-multiply: three times x equals four times one hundred twenty, so three x equals four hundred eighty, and x equals one hundred sixty milliseconds. Keep the same quantity on top of both ratios, requests over time on each side, or the cross-multiplication inverts.

Unit conversion

Chain conversion factors so unwanted units cancel.

Worked example: Convert five gigabytes to megabits. Five gigabytes times one thousand twenty-four megabytes per gigabyte equals five thousand one hundred twenty megabytes. Multiply by eight bits per byte to get forty thousand nine hundred sixty megabits. Writing each factor as a fraction and canceling gigabytes, then megabytes, then bytes keeps you from multiplying when you should divide.

A worked pacing table

Suppose forty questions must be answered in thirty minutes.

So each question gets forty-five seconds on average, and ten questions should take about seven and a half minutes. If you are eleven minutes in at question ten, you are behind pace and should speed up. This rate reasoning, applied to your own clock, is exactly what the test measures.

Averages and combined rates

Worked example: Two servers share a job. Server A finishes a batch in six hours; server B finishes the same batch in three hours. Working together, how long? In one hour A does one-sixth of the batch and B does one-third, which is two-sixths. Together they do three-sixths, or one-half, per hour, so the whole batch takes two hours. The trap is to average six and three to get four and a half hours; rates add, times do not.

A second averaging trap: If you drive to a site at thirty miles per hour and back along the same road at sixty miles per hour, your average speed is not forty-five. Because you spend more time at the slower speed, the true average is the total distance over the total time, which works out to forty miles per hour. Whenever a problem asks for an average rate, combine totals rather than averaging the two rates directly.

Percentages and growth: If a server processes two hundred requests per second now and load is projected to grow by fifteen percent, the new rate is two hundred plus fifteen percent of two hundred, which is two hundred plus thirty, or two hundred thirty requests per second. To reverse a percentage increase you divide rather than subtract: a value that already sits twenty percent above baseline equals the baseline times one and two-tenths, so you divide by one and two-tenths to recover it. A frequent mistake is to subtract the twenty percent instead, which lands below the original, because percentages compound off the current base, not the starting one.

Common traps

• Mixing units. Convert everything to one unit system before calculating.
• Bits and bytes. Divide by eight to go from bits to bytes, and multiply by eight to go the other way.
• Averaging rates directly. Combine work per unit time or total distance over total time, not the two rates.
• Answering in the wrong unit. Re-read the question to confirm whether it wants seconds, minutes, megabytes, or megabits.

Slow down on the setup and the arithmetic becomes routine. Most wrong answers on this section come from a units slip, not from hard math.