2.1 Number Sense, Operations, and Fraction–Decimal–Percent Conversions

2.1 Number Sense, Operations, and Fraction–Decimal–Percent Conversions

Candidate reports single out number sense as the highest-yield mathematics skill on the AdUCET, describing the mathematics portion as broad rather than deep: fractions, decimals, percentages and signed numbers reappear inside geometry items, science items and data-table items, so an examinee who can turn $\frac{5}{8}$ into $62.5\%$ in one beat saves seconds on dozens of questions. Adamson University publishes no content outline, so nobody can tell you how many items any topic receives — but the arithmetic below sits underneath all of them.

1. Order of Operations: GEMDAS

The Philippine K-12 curriculum teaches GEMDAS — Grouping, Exponents, Multiplication and Division, Addition and Subtraction — the same rule taught elsewhere as PEMDAS (parentheses in place of grouping). The detail that costs marks is that the last four letters are two levels, not four:

• Multiplication and division share one level and run left to right: $24 \div 6 \times 2 = 4 \times 2 = 8$, never $24 \div 12 = 2$.
• Addition and subtraction share one level and also run left to right: $10 - 4 + 3 = 9$, not $10 - 7 = 3$.
• A fraction bar, a radical sign and the bars of an absolute value are all invisible grouping symbols — finish the numerator and the denominator before you divide.

Worked Example 1 — signed numbers inside GEMDAS

Evaluate $-3^2 + (-4)^2 \div 8 \times 2 - (5-7)^3$.

1. Grouping. $5 - 7 = -2$.
2. Exponents. $-3^2 = -(3 \cdot 3) = -9$, because the exponent attaches to the 3 alone. $(-4)^2 = +16$, because the parentheses capture the sign. $(-2)^3 = -8$. The line now reads $-9 + 16 \div 8 \times 2 - (-8)$.
3. Multiply and divide, left to right. $16 \div 8 = 2$, then $2 \times 2 = 4$, giving $-9 + 4 - (-8)$.
4. Add and subtract, left to right. $-9 + 4 = -5$, then $-5 - (-8) = -5 + 8 = 3$.

Answer: $3$. Two sign traps are buried in that one line: $-3^2 \neq 9$, and subtracting a negative adds.

2. Divisibility, Prime Factorisation, GCF and LCM

Divisibility tests let you factor and reduce fractions without long division.

Prime factorisation rewrites a number as a product of primes: $84 = 2^2 \cdot 3 \cdot 7$ and $126 = 2 \cdot 3^2 \cdot 7$.

• The greatest common factor (GCF) takes every shared prime to its lowest power: $2^1 \cdot 3^1 \cdot 7^1 = 42$.
• The least common multiple (LCM) takes every prime that appears to its highest power: $2^2 \cdot 3^2 \cdot 7 = 252$.
• Fast check: $\text{GCF} \times \text{LCM}$ equals the product of the two numbers, since $42 \times 252 = 10{,}584 = 84 \times 126$.

Worked Example 2 — deciding between GCF and LCM

A vendor along San Marcelino has 84 pieces of suman and 126 pieces of puto and wants identical baskets with nothing left over. "Largest identical groups" means GCF: 42 baskets, each holding $84 \div 42 = 2$ suman and $126 \div 42 = 3$ puto.

Now a different question. Jeepneys on Route A leave the terminal every 18 minutes and Route B every 24 minutes, both departing at 6:00 AM. "When do the cycles coincide again?" means LCM: $18 = 2 \cdot 3^2$ and $24 = 2^3 \cdot 3$, so the LCM is $2^3 \cdot 3^2 = 72$ minutes — the next simultaneous departure is 7:12 AM. The working rule is short: dividing into groups uses GCF; repeating cycles meeting again uses LCM.

3. Fraction Arithmetic Without a Calculator

• Like denominators: combine numerators only. $\frac{5}{9} + \frac{2}{9} = \frac{7}{9}$.
• Unlike denominators: rewrite both over the least common denominator (LCD), which is the LCM of the denominators.
• Mixed numbers: convert to improper fractions before operating; convert back only if the options are written as mixed numbers.
• Multiplying: cross-cancel first, then multiply straight across.
• Dividing: multiply by the reciprocal of the divisor.

Compute $2\frac{3}{4} - 1\frac{5}{6} + \frac{2}{3}$. The improper forms are $\frac{11}{4}$, $\frac{11}{6}$ and $\frac{2}{3}$; the LCD of 4, 6 and 3 is 12, so the expression becomes $\frac{33}{12} - \frac{22}{12} + \frac{8}{12} = \frac{19}{12} = 1\frac{7}{12}$.

A complex fraction has fractions inside its numerator or denominator. Simplify top and bottom separately, then divide:

$$\frac{\frac{3}{4} + \frac{1}{2}}{\frac{5}{6} - \frac{1}{3}} = \frac{\frac{5}{4}}{\frac{1}{2}} = \frac{5}{4} \times \frac{2}{1} = \frac{10}{4} = \frac{5}{2} = 2.5$$

4. The Conversion Table to Memorise

When a value is not in the table, use the procedures:

1. Fraction to decimal: divide numerator by denominator.
2. Decimal to percent: move the point two places right ($0.045 \to 4.5\%$).
3. Percent to decimal: move the point two places left ($0.5\% \to 0.005$).
4. Decimal to fraction: write over the place-value denominator and reduce ($0.36 = \frac{36}{100} = \frac{9}{25}$).
5. Percent to fraction: write over 100 and reduce ($45\% = \frac{45}{100} = \frac{9}{20}$).
6. Repeating decimal to fraction: put the repeating block over as many 9s as it has digits ($0.\overline{27} = \frac{27}{99} = \frac{3}{11}$).

The table is not trivia — it is a speed tool. Reading $37.5\%$ of 64 as $\frac{3}{8} \times 64$ gives 24 instantly; $16\frac{2}{3}\%$ of 42 is $\frac{42}{6} = 7$; $62.5\%$ of 240 is $\frac{5}{8} \times 240 = 150$. Each of those is a one-line mental computation only if the conversion is automatic.

5. Place Value, Rounding and Estimation

In $3{,}847.652$ the 8 sits in the hundreds place, the 6 in the tenths, the 5 in the hundredths and the 2 in the thousandths. To round, look at the digit immediately to the right of the target place: rounding to the nearest tenth gives $3{,}847.7$ (the hundredths digit is 5, so round up), while rounding to the nearest hundred gives $3{,}800$ (the tens digit is 4, so round down).

Estimation is a scoring tool when four options are far apart. $\frac{7}{8} \times 401 \approx 0.875 \times 400 = 350$, which is close enough to identify the true value $350.875$ without long multiplication.

6. Ordering Mixed Forms

Convert every value to a decimal, then compare digit by digit. Order $0.6$, $\frac{5}{8}$, $58\%$, $\frac{3}{5}$ and $0.\overline{6}$ from least to greatest: the decimals are $0.6$, $0.625$, $0.58$, $0.6$ and $0.666\ldots$, so the order is $58\% < 0.6 = \frac{3}{5} < \frac{5}{8} < 0.\overline{6}$. Note that two of the five are equal — options that force a strict "<" between them are wrong.

For negatives the number line reverses your instinct: $-\frac{3}{4} = -0.75$ lies to the left of $-0.7$, so $-\frac{3}{4} < -0.7$ even though 75 is larger than 70.

Common Traps

• Treating GEMDAS as four ranked steps instead of two shared levels.
• Reading $-5^2$ as $25$; it is $-25$ unless the base is written $(-5)$.
• Adding denominators: $\frac{1}{2} + \frac{1}{3}$ is $\frac{5}{6}$, not $\frac{2}{5}$.
• Converting $0.5\%$ to $0.05$ instead of $0.005$.
• Forgetting that a percent greater than 100 is legal: $125\% = 1.25 = \frac{5}{4}$.
• Comparing $\frac{7}{9}$ and $\frac{5}{7}$ by eye; cross-multiply instead ($7 \times 7 = 49$ against $5 \times 9 = 45$, so $\frac{7}{9}$ is larger).