16.3 Numerical Reasoning, Percentages, and Patterns

16.3 Numerical Reasoning, Percentages, and Patterns

Quick Answer: Some candidate and test-prep write-ups describe numerical items — percentages, simple rates, number patterns — on TAB-style sittings. TSA does not publish numerical reasoning as an official TAB blueprint, and it does not publish a 25–45 minute duration. Your invitation controls the format. Official published numbers you may use are TSA's screening volumes: approximately 3.3 million carry-on bags and 1.3 million checked bags daily. Any lane counts, 15% additional-screening rates, or bags-per-hour figures in this section are labeled practice numbers, not TSA statistics. Work the arithmetic in writing so Attention to Detail includes the digits.

Official purpose versus practice arithmetic

TAB's official purpose remains x-ray interpretation aptitude. Competencies include Visual Observation, Attention to Detail, and Critical Thinking. Spatial and image tasks are the obvious match. Numbers still show up in TSO work: how many bags are on a belt, how much of a practice hour is additional screening, simple counts of bins. That does not convert a practice percentage item into an official TSA section.

The invitation controls the format. If you see a percentage item, compute it. If you do not, you have not been cheated by this chapter — you practiced Attention to Detail with digits, which the JOA already names. Folding, matching, trail-making, a 25–45 minute clock, and numerical modules all sit in the candidate-reported column of the section 16.1 table. Do not promote them to TSA.gov.

Official volume (TSA-published). TSA's public screening materials state that TSA screens approximately 3.3 million carry-on bags daily and approximately 1.3 million checked bags daily. Those two figures are the only bag-volume statistics this section treats as official. Combined, 3.3 million + 1.3 million = 4.6 million bags of those two types per day.

Carry-on share of that combined total: 3.3 ÷ 4.6. Compute it. 4.6 × 0.7 = 3.22. Remainder 0.08. 0.08 ÷ 4.6 ≈ 0.017. So about 71.7% carry-on. Checked share: 1.3 ÷ 4.6. 4.6 × 0.28 = 1.288. Remainder 0.012, which is about 0.3 percentage points, so about 28.3% checked. Check: 71.7% + 28.3% = 100%. Those percentages are derived from TSA's published daily volumes. They are not a TAB answer key and they are not a lane mix.

Practice percentages (not TSA stats)

The following checkpoint is invented for practice. It is not a TSA airport, not a published additional-screening rate, and not a TAB item.

Practice checkpoint Alpha processes 800 carry-on bags in one hour across 4 lanes.

• Bags per lane per hour: 800 ÷ 4 = 200 bags per lane per hour.
• Bags per lane per minute: 200 ÷ 60 = 10/3 ≈ 3.33 bags per minute (exactly 3 and 1/3).
• If 15% of the 800 bags require additional screening (practice rate only): 0.15 × 800. 0.10 × 800 = 80. 0.05 × 800 = 40. 80 + 40 = 120 bags. The remaining 85%: 0.85 × 800. 0.80 × 800 = 640. 0.05 × 800 = 40. 640 + 40 = 680 bags that the practice story treats as resolved on the first image. Check: 120 + 680 = 800.
• If 3 of 4 lanes are standard 2-D X-ray and 1 of 4 is CT, the CT share of lanes is 1/4 = 25%, and the standard share is 3/4 = 75%. That is a practice lane mix, not TSA's national equipment inventory.

Practice checkpoint Bravo handles 12,000 carry-on bags in a day (practice, not a TSA-published airport total). 1,800 of those bags are pulled for additional screening (practice). Percentage: 1,800 ÷ 12,000. Divide numerator and denominator by 600: 3 ÷ 20 = 0.15 = 15%. If the next day the additional-screening count is 2,400 of 12,000: 2,400 ÷ 12,000 = 24 ÷ 120 = 0.20 = 20%. The increase is 20% − 15% = 5 percentage points. That is not the same as a 5% increase of the original 15%, which would be 0.05 × 15% = 0.75 percentage points. Percentage points versus percent-of-a-percent is an Attention to Detail trap. If an item asks how much the rate rose, read whether it wants the difference in percentage points (5) or the relative change (5 ÷ 15 = 1/3 ≈ 33.3% relative increase of the rate). Those are different numbers.

Practice: what percent is Bravo of the official daily carry-on volume? Official daily carry-on volume is about 3.3 million. 12,000 / 3,300,000 = 12 / 3,300 = 4 / 1,100. 4 ÷ 1,100 = 0.003636… ≈ 0.364%. A single practice airport's 12,000 bags are a fraction of one percent of the published national carry-on total. Do not treat a practice lane as if it were TSA's 3.3 million figure, and do not treat 3.3 million as if it were one checkpoint's hourly throughput.

Rate problems

Rates are quantity per time. Bags per hour, bags per minute, minutes per bag. Write the units. If the units disappear, the rate is no longer a rate.

Practice: 200 bags per hour per lane. How many bags in 45 minutes at that rate? 45 minutes is 45/60 = 0.75 hour. 200 × 0.75 = 150 bags. Alternatively: 200 bags / 60 minutes = 10/3 bags per minute; (10/3) × 45 = 450/3 = 150. Same result, two routes. If they disagree, an arithmetic error happened; do not pick the prettier number.

Practice: A lane must move 180 bags in 90 minutes. Required rate: 180 ÷ 90 = 2 bags per minute, which is 2 × 60 = 120 bags per hour. That is slower than Alpha's practice 200 bags per hour. The comparison is only as good as the practice labels. Neither figure is a TSA-published checkpoint standard.

Practice: Four lanes each at 200 bags/hour for 3 hours: 4 × 200 × 3 = 800 × 3 = 2,400 bags. If 10% (practice) need additional screening: 0.10 × 2,400 = 240. If 15% instead: 0.15 × 2,400 = 360. The difference between those two practice rates is 360 − 240 = 120 bags, which happens to match Alpha's 15%-of-800 figure for a different reason (different whole). Always go back to percent × this item's whole. Do not recycle a previous 120 without checking the whole.

Do not divide 3.3 million by 4 and call it an official lane rate. 3.3 million is a national daily carry-on volume. Four is Alpha's practice lane count. Mixing them produces a fiction that looks precise and is source-dishonest.

Simple patterns

Number-pattern items ask for the next value or the rule. Work them on paper. They are not official TSA threat formulas, and finding a pattern is not SSI.

Multiply-by-constant: 2, 6, 18, 54, … Each term × 3. Next: 54 × 3 = 162.

Double: 7, 14, 28, 56, … Next: 56 × 2 = 112.

Squares: 4, 9, 16, 25, 36, … which is 2², 3², 4², 5², 6². Next: 7² = 49.

Add previous two (Fibonacci-like): 5, 8, 13, 21, … because 5 + 8 = 13 and 8 + 13 = 21. Next: 13 + 21 = 34. Always test add the previous two before inventing a complicated alternating rule. Differences +3, +5, +8 looked messy; the sum rule is clean.

Alternating operations: 3, 6, 5, 10, 9, 18, 17, … Pattern: ×2, then −1, then ×2, then −1. After 17, ×2 = 34. Confirm one more step backward: 18 − 1 = 17, 9 × 2 = 18, 10 − 1 = 9. The rule holds.

Patterns on a hiring test are Attention to Detail with sequences. Write the candidate rule, test it on every given term, then apply it once. A rule that fits only the last two terms is a guess.

Putting numbers next to images without mixing sources

A candidate who is good at folding but treats 15% of 800 as 15, or who quotes 25–45 minutes as TSA.gov, still fails the honesty and the arithmetic. Critical Thinking is choosing the right operation: percent of a whole is multiply; percent that a part is of a whole is divide; a rate needs a time unit; a pattern needs a rule that survives every term.

Worked wrap-up using only labeled numbers. Official: 3.3 million carry-on per day and 1.3 million checked per day, combined 4.6 million, about 71.7% carry-on. Practice: checkpoint Alpha, 800 bags/hour, 4 lanes, 15% additional screening → 200 bags/lane/hour and 120 additional-screening bags that hour; 45 minutes at 200 bags/hour is 150 bags. Pattern practice: next square after 36 is 49; next Fibonacci-like term after 5, 8, 13, 21 is 34. Source status: folding, matching, trail-making, 25–45 minutes, and these lane percentages remain candidate-reported or practice, not a TSA.gov TAB blueprint. Official competencies are still Visual Observation, Attention to Detail, and Critical Thinking. Official TAB purpose is still x-ray interpretation aptitude. Official CT fact is still a 3-D image rotated 360 degrees. The invitation controls the format.

Traps

• Presenting practice 15% additional-screening rates, 800 bags/hour, or 200 bags/lane/hour as TSA statistics.
• Confusing 3.3 million daily national carry-on with a single-lane hourly rate.
• Treating 5 percentage points as 5% of 15%.
• Calling numerical items an official TAB section, or quoting 25–45 minutes as published.
• Skipping the written calculation and guessing from how large the numbers look.
• Recycling an answer from a previous practice whole (the 120 from 0.15 × 800) onto a new whole without multiplying again.

If the invitation has no numerical items, you still gained the habit of checking digits — the same Attention to Detail the JOA names for an image-interpretation aptitude test. If the invitation does include arithmetic, you already worked the operations instead of memorizing a vendor worksheet.