5.4 Number Series, Age Problems & Capacity Checks

5.4 Number Series, Age Problems & Capacity Checks

FPSI's sample Mathematical Reasoning items include three short problem types that are easy points once you know the method: a number series written as fractions, an age problem, and a weight-capacity problem. All three use only basic arithmetic, but each has a predictable trap.

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Number Series

A series item shows several terms and asks which number comes next. The pattern is always a rule you can state in a few words, such as "add 4," "double and add 1," or "decrease by one-quarter."

A Four-Step Method

1. Check the differences. Subtract each term from the next. If the differences are equal, add that difference again.
2. Check second differences. If the differences change by a constant amount (for example 2, 4, 6, 8), the next difference continues that pattern.
3. Check ratios. If each term is a multiple of the last (×2, ×3, ÷2), continue the ratio.
4. Check two-step and alternating rules. Some series combine operations (×2 then +1) or alternate between two rules or two interleaved series.

Series Written as Fractions

FPSI's sample writes its series as fractions that are not in lowest terms, which hides a simple pattern. Simplify every term first, or convert to decimals.

Example: 30/30, 16/20, 9/15, ?

• 30/30 = 1
• 16/20 = 4/5
• 9/15 = 3/5

The values fall by one-fifth each time (1, 4/5, 3/5), so the next term is 2/5. Converting to decimals tells the same story: 1.0, 0.8, 0.6, so the next value is 0.4. An answer choice might show it in another form, such as 4/10 or 6/15, so compare values rather than appearances. A common trap repeats the last fraction's form or continues a pattern in the raw numerators instead of the values.

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Checking a Series Answer

Before you mark an answer, test the rule on every pair of neighboring terms, not just the first two. In 2, 4, 8, ..., both "double" (next 16) and "add 2, then 4, then 6" (next 14) fit the first three terms, so a well-written item gives at least four terms or offers only one option that fits. If two rules seem to fit, choose the one that matches all the terms shown and appears among the options. When a series alternates, split it into two interleaved series: 3, 20, 6, 17, 9, 14, ... is 3, 6, 9 (add 3) woven together with 20, 17, 14 (subtract 3), so the next term is 12.

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Age Problems

Age problems rest on one fact: everyone ages at the same rate, so the difference between two people's ages never changes, while the ratio does.

Example: An officer is 42, and her son is one-third her age. How old will the son be when the officer is 54?

1. Son's current age: $42 \div 3 = 14$.
2. Years that will pass: $54 - 42 = 12$.
3. Son's future age: $14 + 12 = 26$.

The trap answer keeps the ratio fixed and divides 54 by 3 to get 18. Ratios between ages change over time; differences do not. (The son is 28 years younger now and will still be 28 years younger.)

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Capacity and Limit Problems

A capacity problem gives a limit (weight, seats, hours, budget) and several amounts, then asks whether adding one more amount stays within the limit and by how much.

1. Add every amount that counts toward the limit, including the new one.
2. Compare the total with the limit.
3. Report the difference the question asks for: "exceeded by" means total minus limit; "remaining capacity" means limit minus total.

Example: A transport van is rated for 1,500 pounds. Four officers weighing 182, 205, 168, and 194 pounds are aboard with 640 pounds of equipment. Can a 210-pound passenger be added?

• Officers: $182 + 205 + 168 + 194 = 749$ pounds.
• Officers plus equipment: $749 + 640 = 1{,}389$ pounds.
• Adding the passenger: $1{,}389 + 210 = 1{,}599$ pounds.
• Over the limit by: $1{,}599 - 1{,}500 = 99$ pounds.

The answer is no; the limit would be exceeded by 99 pounds. Common traps forget the equipment (giving 959 pounds, well under the limit) or subtract in the wrong order.

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Translating Words into Operations

Keywords are a starting point, not a substitute for reading. Always ask what quantity the final sentence requests, and check that your answer has the right units.