Fractions, Decimals & Order of Operations: The Arithmetic Foundation

Before special triangles and quadratics, there's arithmetic — and shaky arithmetic quietly costs points on every test. A surprising share of wrong answers aren't from not knowing the concept; they're from a fraction added wrong or the order of operations slipped. Solid fluency with fractions, decimals, and the order of operations is the foundation everything else is built on, and it's very much worth shoring up.

This guide reviews the arithmetic essentials, with worked and practice problems matched to real test difficulty at Northside Tutoring.

Why the Foundations Matter

Arithmetic fluency speeds up every other topic and prevents careless errors. It matters most on the SSAT (where these are tested directly and calculators aren't used) but underpins the SAT and ACT too. Getting it automatic frees your attention for the actual problem.

Order of Operations (PEMDAS)

When an expression mixes operations, you must do them in the right order, remembered as PEMDAS:

Order	Operation
1. P	Parentheses (innermost first)
2. E	Exponents
3. MD	Multiply & Divide, left to right
4. AS	Add & Subtract, left to right

So 2 + 3 × 4 = 2 + 12 = 14 (not 20) — multiplication comes before addition.

Operations With Fractions

The rules, briefly: to add or subtract, get a common denominator first, then combine numerators (⅔ + ¼ = 8/12 + 3/12 = 11/12). To multiply, go straight across (⅔ × ¼ = 2/12 = ⅙). To divide, flip the second fraction and multiply (⅔ ÷ ¼ = ⅔ × 4 = 8/3).

Operations With Decimals

To add or subtract decimals, line up the decimal points. To multiply, ignore the points, multiply, then place the decimal so the answer has as many decimal places as the two factors combined (0.2 × 0.3 = 0.06). To divide, shift both decimals to make the divisor whole.

Converting Between Forms

Fractions, decimals, and percents are three ways to write the same value — convert freely. A fraction becomes a decimal by dividing (¾ = 3 ÷ 4 = 0.75); a decimal becomes a percent by moving the point two places (0.75 = 75%). Knowing common equivalents (½ = 0.5 = 50%, ¼ = 0.25 = 25%) saves time.

Avoiding Careless Errors

Arithmetic slips are the most common avoidable mistakes on any test. Slow down a touch on the computation, write out steps rather than doing everything mentally, and on the no-calculator SSAT especially, double-check fraction and decimal work. The points lost to careless errors are the easiest ones to win back.

Where You'll See This — Test by Test

Arithmetic is the foundation of every test. It's most directly tested (and calculator-free) on the SSAT, but solid fractions, decimals, and order-of-operations fluency prevents careless errors on the SAT and ACT too.

Watch the Lesson

Sometimes a diagram needs a voice. In the short video below, one of our Northside tutors walks through the core idea and works through test-style problems in real time.

Worked Example Problems

These problems are calibrated to the difficulty you'll actually see on test day. Try each one before opening the solution.

Common Mistakes to Avoid

Practice Problems — You Try

Three problems below. Work each before checking the solution.

The Northside Method — How We Teach This 1-on-1

Reading a blog is a great starting point. But there's a meaningful gap between understanding a concept and reflexively applying it under timed conditions. That gap is exactly what our tutors close.

Every Northside student works through a four-step framework:

1. Assessment. We diagnose which specific skills are slowing your student down — not just whether they "get it" in the abstract.
2. Perfect-match coach. We pair them with an elite tutor (we accept only the top 1% of applicants) whose teaching style fits how your student actually learns.
3. Bespoke plan. A roadmap built around your student's target score, target timeline, and current pacing data.
4. Data-driven adjustment. Every session ends with a check on whether the student's accuracy and speed are moving in the right direction.

And if a student meets all eligibility requirements but doesn't hit the defined score improvement? We provide 5 additional hours of cohort learning at no cost. That's the Northside guarantee — built on 25 years of measured outcomes.