5.1 Fundamental Operations

5.1 Fundamental Operations on NMAT Quantitative

The Center for Educational Measurement (CEM) NMAT Quantitative subtest is a 30-item block with a recommended ~40 minutes. Official item families include fundamental operations (direct computation), problem solving, and data interpretation. Together they measure how well you organize and apply basic mathematical concepts and quantitative reasoning — not advanced calculus.

This section trains the first family: fast, correct arithmetic without a calculator. On test day you will not outsource signed numbers, fraction arithmetic, percent, roots, or order of operations to a device. Fluency here frees time for multi-step word problems and table reads later in the same subtest.

Quick frame: A fundamental-operations item is usually "compute this exact value." If you are rewriting the whole problem as algebra, you may be overcomplicating a direct evaluation. If you skip checking signs or the percent base, you will land on a distractor built for that exact slip.

What "fundamental operations" means on NMAT

Expect items that combine:

• Integers with addition, subtraction, multiplication, division, and signs
• Fractions and mixed numbers (simplify, convert, operate)
• Decimals and place-value discipline
• Percentages (of a base, increase/decrease, successive percents)
• Order of operations (parentheses, exponents, multiply/divide left-to-right, add/subtract left-to-right)
• Exponents and roots at high-school fluency (squares, cubes, square roots of perfect squares, simple fractional powers)
• Ratios and proportions stated as computations (not always full word stories)

Medical or lab-flavored numbers appear because the NMAT audience is pre-med, but the test is still arithmetic. Treat "mg," "mL," and "bpm" as labels that demand unit consistency — they do not change the math rules.

Order of operations (non-negotiable skeleton)

Use a single consistent order:

1. Grouping symbols: parentheses, brackets, fraction bars as grouping
2. Exponents and roots
3. Multiplication and division left to right
4. Addition and subtraction left to right

Worked computation A
Evaluate: ( 8 + 12 \div 4 \times 3^2 - 5 )

• Exponent first: ( 3^2 = 9 ) → ( 8 + 12 \div 4 \times 9 - 5 )
• Multiply/divide L→R: ( 12 \div 4 = 3 ), then ( 3 \times 9 = 27 ) → ( 8 + 27 - 5 )
• Add/subtract L→R: ( 35 - 5 = 30 )

Trap: Doing ( 8 + 12 = 20 ) first, or treating ( 12 \div 4 \times 9 ) as ( 12 \div 36 ). Both produce options you will see in multi-choice sets.

Worked computation B (fraction bar as grouping)
( \dfrac{15 - 3}{2 + 4} = \dfrac{12}{6} = 2 )
The bar groups numerator and denominator separately before dividing.

Integers and sign control

Signed arithmetic is where many timed misses hide.

Trap cluster: ( -3^2 ) vs ( (-3)^2 ). On paper, write parentheses when the base is negative. NMAT distractors love ( -9 ) vs ( 9 ).

Worked computation C
( (-12) \div 3 + (-2)(-5) - 7 )
( = -4 + 10 - 7 = -1 )

Fraction fluency without a calculator

You need speed on:

• Simplify before multiplying when possible
• Common denominators for add/subtract
• Invert and multiply for division
• Mixed ↔ improper conversions
• Compare fractions by cross-multiply or benchmark (( 1/2 ), ( 1/3 ), ( 1/4 ), ( 2/3 ), ( 3/4 ))

Worked computation D
( \dfrac{3}{8} + \dfrac{5}{12} )
LCD of 8 and 12 is 24: ( \dfrac{9}{24} + \dfrac{10}{24} = \dfrac{19}{24} )

Worked computation E
( \dfrac{2}{5} \times \dfrac{15}{8} = \dfrac{2 \times 15}{5 \times 8} ). Cancel 5: ( \dfrac{2 \times 3}{1 \times 8} = \dfrac{6}{8} = \dfrac{3}{4} )

Worked computation F
( \dfrac{7}{9} \div \dfrac{14}{15} = \dfrac{7}{9} \times \dfrac{15}{14} = \dfrac{7 \times 15}{9 \times 14} ). Cancel 7 and 3 from 9 and 15: ( \dfrac{1 \times 5}{3 \times 2} = \dfrac{5}{6} )

Medical-context framing (original, arithmetic only)
A vial is labeled so that ( \dfrac{2}{5} ) of the volume is active ingredient and a nurse draws ( \dfrac{3}{4} ) of the vial. Fraction of a full vial that is active ingredient drawn: ( \dfrac{2}{5} \times \dfrac{3}{4} = \dfrac{6}{20} = \dfrac{3}{10} ). The story is clinical flavor; the operation is product of fractions.

Decimals and place value

• Align decimal points for add/subtract
• Count total decimal places for multiply
• For divide, shift the decimal in divisor and dividend the same number of places
• Prefer converting recurring or messy decimals to fractions when the options are fractional

Worked computation G
( 0.25 \times 0.8 = 0.2 ) (or ( 1/4 \times 4/5 = 1/5 = 0.2 ))

Worked computation H
( 1.2 \div 0.03 ): rewrite as ( 120 \div 3 = 40 )

Percent: "of" vs "change" (highest-yield trap)

Percent of a base
( p% ) of ( B = \dfrac{p}{100} \times B )

Example: ( 15% ) of 240 = ( 0.15 \times 240 = 36 )

Percent change
( \dfrac{\text{new} - \text{old}}{\text{old}} \times 100% )
Increase uses positive; decrease is negative or stated as "decreased by."

Example: A count rises from 80 to 100.
Change = ( \dfrac{20}{80} = 0.25 = 25% ) increase — not "20%" from ( 20/100 ).

Percent of vs points
If a rate goes from 40% to 50%, that is a 10 percentage-point rise, but a 25% relative increase (( 10/40 )). Multi-choice stems that say "increased by 25%" vs "increased by 10%" exploit this confusion.

Successive percents do not add
A price up 10% then down 10% is not back to original:
( 1.10 \times 0.90 = 0.99 ) → 1% net decrease.

Worked computation I (medical-style rate)
A resting heart rate of 72 bpm rises by 25% during a stair test. New rate: ( 72 \times 1.25 = 90 ) bpm.
If someone wrongly does ( 72 + 25 = 97 ), they confuse percent with raw points.

Worked computation J (dose % of daily total)
A patient is prescribed 1,200 mg of a drug per day. Morning dose is 35% of the daily total. Morning amount: ( 0.35 \times 1200 = 420 ) mg.

Percent–fraction–decimal equivalents (commit to memory)

Benchmark habit: 10% = move decimal one place left; 5% = half of 10%; 1% = move two places. Build 15% as 10% + 5%, 35% as 25% + 10%, etc.

Exponents and roots (fluency set)

• ( a^m \cdot a^n = a^{m+n} ); ( \dfrac{a^m}{a^n} = a^{m-n} ) (( a \neq 0 ))
• ( (a^m)^n = a^{mn} )
• ( a^0 = 1 ) for ( a \neq 0 )
• ( a^{-n} = \dfrac{1}{a^n} )
• Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
• Perfect cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000
• ( \sqrt{ab} = \sqrt{a}\sqrt{b} ) for non-negative values as used on school tests; simplify radicals: ( \sqrt{50} = 5\sqrt{2} )

Worked computation K
( 2^5 \cdot 2^{-2} = 2^{3} = 8 )

Worked computation L
( \sqrt{144} + \sqrt{0.16} = 12 + 0.4 = 12.4 )
(Do not invent ( \sqrt{a+b} = \sqrt{a} + \sqrt{b} ) — that identity is false.)

Ratios and proportional scaling

A ratio ( a:b ) is the fraction ( a/b ) with order fixed. Scaling: if ( a:b = 3:5 ) and ( a = 12 ), then ( 12/b = 3/5 ) → ( b = 20 ).

Worked computation M (dilution framing)
A solution mixes concentrate to water in ratio 2:5. If concentrate is 80 mL, water is ( 80 \times \dfrac{5}{2} = 200 ) mL. Total mixture = 280 mL. Concentrate fraction of total = ( 2/(2+5) = 2/7 ).

Mental-math routes that protect the 40-minute block

1. Factor before expand: ( 15 \times 28 = 15 \times (30 - 2) = 450 - 30 = 420 )
2. Compatible numbers: ( 25 \times 48 = 25 \times 50 - 25 \times 2 = 1250 - 50 = 1200 )
3. Fraction benchmarks: Is ( 17/40 ) more than half? Half of 40 is 20, so no.
4. Cancel early in multiplications of fractions.
5. Estimate last digit or magnitude before committing: ( 19.8 \times 5.1 \approx 20 \times 5 = 100 ); exact near 100.98.
6. Rewrite percent as fraction when the base is a multiple of the denominator (25% of 80 = 1/4 of 80 = 20).

Common error traps (checklist)

Mini drill set (study only — original)

Compute, then check with a reverse operation or estimation:

1. ( 48 - 3(5 - 9)^2 )
   Inside: ( 5-9=-4 ); square 16; ( 3\times16=48 ); ( 48-48=0 ).
2. ( \dfrac{5}{6} - \dfrac{1}{4} = \dfrac{10}{12} - \dfrac{3}{12} = \dfrac{7}{12} )
3. 12.5% of 640 = ( \dfrac{1}{8} \times 640 = 80 )
4. From 200 mg to 150 mg: percent decrease = ( 50/200 = 25% )
5. Ratio 4:7, second term 35 → first term = ( 35 \times 4/7 = 20 )

Section checkpoint

You are ready for problem-solving items when you can: (1) evaluate mixed expressions with signs and exponents without reordering carelessly, (2) convert among fraction, decimal, and percent in under five seconds for the tabled values, (3) distinguish percent of from percent change out loud, and (4) finish a short arithmetic item in well under a minute so multi-step word problems have room on the Quantitative clock.