Pre-Algebra Class 5: Equivalent Fractions, Comparing & Converting to Decimals

Class 5 is where students stop thinking of fractions as isolated shapes on a page and start treating them as numbers — numbers that can be rewritten, compared, and converted to decimals. Every question on the SHSAT, SAT, and Regents that involves fractions rests on the three moves in this class.

What a fraction really is

Before the arithmetic, the vocabulary. The fraction a/b (with b ≠ 0) has three equally valid meanings that show up on tests:

1. Part of a whole. Cut the whole into b equal pieces; take a of them. 3/4 of a pizza means slice the pizza into 4 equal pieces and take 3.
2. A division. a/b = a ÷ b. The fraction bar is literally a division sign. 3/4 = 3 ÷ 4 = 0.75.
3. A point on the number line. Between whole numbers 0 and 1 you can slide to any exact spot: 1/2 sits halfway; 3/4 sits three-quarters of the way; 5/4 is one whole plus one more quarter, so it lives between 1 and 2.

The numerator (top) counts pieces. The denominator (bottom) names the size of each piece. Change the denominator and you're changing the unit; change the numerator and you're changing the count. Every technique in this class comes back to keeping the size of a piece coordinated with how many of them you have.

The three moves you must know cold

Every fraction question in Class 5 uses one of these three ideas. Master them and the arithmetic slows down while the confidence speeds up.

Move 1 — Building an equivalent fraction (and why it works)

Multiplying both the numerator and the denominator by the same nonzero number k does not change the value. In symbols:

a/b = (a · k) / (b · k)

Why. You are really multiplying by k/k, which equals 1. Multiplying anything by 1 leaves it unchanged. So 3/4 · 5/5 = 15/20, and 3/4 = 15/20 as points on the number line. Picture it: cutting each of the 4 original pieces into 5 smaller pieces gives 20 tiny pieces total, and the 3 original shaded pieces become 15 tiny shaded pieces — same amount, just finer slicing.

The same reasoning runs in reverse: dividing top and bottom by a common factor k also keeps the value fixed. That reverse move is called simplifying or reducing.

Move 2 — Simplifying to lowest terms with the GCF

A fraction is in lowest terms (or simplest form) when the numerator and denominator share no common factor other than 1. To get there in one step, divide both by their Greatest Common Factor (GCF) — the largest whole number that divides both.

Two ways to find the GCF:

• List factors. Best for small numbers. Factors of 18: 1, 2, 3, 6, 9, 18. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. Biggest shared factor is 6. So 18/24 = 3/4.
• Prime factorization. Best for larger numbers. Write each as a product of primes and multiply the shared ones. 36 = 2² · 3². 60 = 2² · 3 · 5. Shared: 2² · 3 = 12. So 36/60 = 3/5.

Why bother simplifying? Two reasons. First, lowest terms is the standard answer form on almost every test — graders take off credit for an un-simplified fraction. Second, it makes the number easier to recognize: 21/28 hides the fact that it equals 3/4, which is a benchmark you already know as 0.75 or 75%.

Move 3 — Comparing fractions

You have three tools. Pick whichever is fastest for the two fractions in front of you.

1. Same denominator? The bigger numerator wins. 3/8 < 5/8 because the pieces are the same size, so you're just counting pieces.
2. Same numerator? The smaller denominator wins — fewer, larger pieces beats more, smaller pieces. 2/5 > 2/7 because fifths are bigger than sevenths.
3. Nothing in common? Rewrite with a common denominator, or cross-multiply.

Cross-multiplication, explained. To compare a/b and c/d (both denominators positive), compute a · d and c · b. Whichever cross-product is bigger sits over the bigger fraction. The trick works because it's really just "give them a common denominator bd without writing it out": a/b becomes ad/bd and c/d becomes bc/bd. Same denominator, so compare numerators — that's ad vs. bc.

a/b > c/d  ⇔  a · d > c · b

Warning. Cross-multiplication only compares. It does not give you a new fraction and it does not add or subtract fractions — those still need a common denominator.

Benchmark fractions. When you need to sanity-check a comparison in your head, anchor to fractions you know cold:

Ask "is this fraction bigger or smaller than 1/2?" and half of comparison problems answer themselves. 7/13 > 1/2 because 7 > half of 13; 5/12 < 1/2 because 5 < half of 12.

Converting a fraction to a decimal

To turn a fraction into a decimal, divide the numerator by the denominator. Two shortcuts and one long-division fallback:

• Shortcut A — denominator is 10, 100, or 1000. Read the digits directly. 47/100 = 0.47. 9/10 = 0.9. 123/1000 = 0.123.
• Shortcut B — rewrite the denominator to a power of 10. If the denominator divides some power of 10, scale it up. 3/4: multiply by 25/25 to get 75/100 = 0.75. 7/25: multiply by 4/4 to get 28/100 = 0.28. 3/8: multiply by 125/125 to get 375/1000 = 0.375.
• Fallback — long division. When the shortcuts fail, just divide. For 3/8: 3 ÷ 8 = 0.375. For 1/3: 1 ÷ 3 = 0.333… The division either ends (terminates) or one of the remainders repeats and the digits start cycling.

Terminating vs. repeating — and why

Whether a fraction terminates or repeats depends only on the denominator in lowest terms. Look at its prime factorization:

Why the rule works. A terminating decimal is really a fraction with denominator 10, 100, 1000, and so on — that is, a power of 10. And every power of 10 factors as 2s and 5s: 10 = 2 · 5, 100 = 2² · 5², 1000 = 2³ · 5³. If your denominator's primes are already 2s and 5s, you can scale it up to a power of 10 by multiplying in missing 2s or 5s. If the denominator has a 3, a 7, or any other prime, no power of 10 will ever contain it — the decimal must repeat.

Notation for repeating decimals. Draw a bar over the block that repeats. 1/3 = 0.3 means 0.3333… 4/11 = 0.36 means 0.363636… The repeating block can be 1, 2, or more digits long — it will always be shorter than the denominator.

Going the other way — decimal to fraction

For a terminating decimal, read the digits over the right power of 10, then simplify.

• 0.4 = 4/10 = 2/5
• 0.25 = 25/100 = 1/4
• 0.375 = 375/1000 = 3/8

The number of digits after the decimal point tells you the number of zeros in the denominator: one digit → over 10, two digits → over 100, three digits → over 1000.

Two rules students always forget

Where this class shows up later

Every technique in Class 5 is a foundation for the next few classes and for every state test. Some previews:

• Class 6 (four operations on fractions). Every addition or subtraction of fractions needs equivalent fractions with a common denominator — that's Move 1 from this class.
• Class 8–10 (ratios, proportions, percent). A percent is just a fraction with denominator 100, and a proportion (a/b = c/d) is solved by exactly the cross-multiplication you learn here.
• Algebra 1. Simplifying a rational expression like (x² − 1) / (x + 1) is the same GCF move as simplifying 18/24 — you cancel a common factor from top and bottom.
• SHSAT and SAT. Both tests love fraction-vs-decimal comparison questions where the fast student converts everything to decimals and the slow student tries to find a common denominator by hand.

The 25 questions below march from pure equivalence drill through simplifying, comparing, and converting to decimals, and finish with real-life word problems where fractions and decimals appear side by side.

25 practice questions with hidden answers

About the Young Fermats Pre-Algebra course

Class 5 is one of 48 rolling classes in our Young Fermats Pre-Algebra program for grade 5–6 students (ages 10–12). The full arc runs from integers on the number line through systems of linear equations and probability — the direct on-ramp to Algebra 1 in 7th or 8th grade. Classes are small-group (max 6 students), 120 minutes per week, and students can start any Monday because the syllabus is rolling.

Every new family starts with a free 60-minute evaluation in our 226 W 79th Street classroom plus a written diagnostic delivered within 48 hours. The diagnostic maps your child's current level against the readiness signals for Pre-Algebra and Algebra 1 — and gives you a clear next step whether you enroll or not.

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