MS2 Algebra I — Rational Numbers Quiz

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📌 KEY FORMULAS — Quick Reference

Rational = a/b, b≠0

terminating → denominator: 2ⁿ·5ᵐ only

repeating → all others

ℤ ⊂ ℚ ⊂ ℝ

x² = 2 → x = √2 (irrational!)

✦ SECTION 1 · Definitions & Basic Concepts ✦

RATIONAL = RATIOnal → it's a ratio!
Any number you can write as top ÷ bottom (and bottom ≠ 0) ✅

★☆☆☆☆ EASY

Q1 of 20

What is the definition of a rational number?

THINK ABOUT IT

The word "rational" comes from "ratio" — a comparison of two quantities.

💡 EXPLANATION

Rational numbers are written as a/b where a and b are both integers, and b ≠ 0 (we can never divide by zero!). Examples: ½, -3, 0, 7/4. The key word is integers — both top and bottom must be whole numbers or their negatives.

★☆☆☆☆ EASY

Q2 of 20

Which of the following is NOT a rational number?

💡 EXPLANATION

√2 = 1.41421356... — it goes on forever with NO repeating pattern. You can't write it as a/b. This makes it irrational.
• −7 = −7/1 ✅ rational
• 0.333... = 1/3 ✅ rational (repeating!)
• 0 = 0/1 ✅ rational

★☆☆☆☆ EASY

Q3 of 20

Express 1340 as a decimal. What do you get?

HINT — Long Division

13 ÷ 40 = ? Try: 130 ÷ 40 = 3 remainder 10 → 100 ÷ 40 = 2 remainder 20 → ...

💡 EXPLANATION

13 ÷ 40: Multiply both by 2.5 → 32.5/100 = 0.325. Or long divide:
13.000 ÷ 40 = 0.325 exactly — a terminating decimal!
Why terminating? Because 40 = 2³ × 5, which only has factors of 2 and 5. ✅

★★☆☆☆ BASIC

Q4 of 20

Express 638 as a decimal.

💡 EXPLANATION

63 ÷ 8: 8 goes into 63 seven times (7×8=56), remainder 7.
Then 70 ÷ 8 = 8 r6 → 60 ÷ 8 = 7 r4 → 40 ÷ 8 = 5 exactly.
Answer: 7.875 — terminating because 8 = 2³ (only factor of 2).

✦ SECTION 2 · Terminating vs. Repeating Decimals ✦

TRICK Terminating? → Check denominator (in lowest terms).
Only 2s and 5s? → Terminates ✅ Anything else? → Repeats 🔁

★★☆☆☆ BASIC

Q5 of 20

Express 137 as a decimal. What type is it?

NOTE

7 = 7 (prime, not 2 or 5) → What does this tell you?

💡 EXPLANATION

13 ÷ 7 = 1.857142857142... — the block "857142" repeats forever!
Written as 1.857142 (bar over the repeating block).
Denominator 7 has a prime factor other than 2 or 5 → always repeating. The period (length of repeating block) is at most 6 for denominator 7.

★★☆☆☆ BASIC

Q6 of 20

Which fraction gives a terminating decimal?

💡 EXPLANATION

Check each denominator (in lowest terms):
• 3 → has factor 3 → repeating
• 7 → has factor 7 → repeating
• 16 = 2⁴ → only 2s → TERMINATING ✅ (3/16 = 0.1875)
• 12 = 2²×3 → has factor 3 → repeating

★★★☆☆ MEDIUM

Q7 of 20

Convert 0.6̄ (= 0.666...) to a fraction. What is the correct fraction?

STRATEGY — The Algebra Trick

Let x = 0.666... → 10x = 6.666... → subtract: 9x = 6 → x = ?

💡 EXPLANATION

Let x = 0.666...
10x = 6.666...
10x − x = 6.666... − 0.666...
9x = 6 → x = 6/9 = 2/3

⚠️ Always simplify! 6/9 ÷ 3/3 = 2/3 ✅

★★★☆☆ MEDIUM

Q8 of 20

Convert 0.27̄ (= 0.2777...) to a fraction.

STRATEGY

1 digit repeats → multiply by 10, then by 100. Subtract carefully!
Let x = 0.2777... → 10x = 2.777... → 100x = 27.777...

💡 EXPLANATION

x = 0.2777...
100x = 27.777...
10x = 2.777...
100x − 10x = 27.777... − 2.777...
90x = 25 → x = 25/90 = 5/18 ✅

⚠️ Common mistake: subtracting x instead of 10x → get wrong denominator!

✦ SECTION 3 · Number System & Classification ✦

ORDER Natural ⊂ Whole ⊂ Integer ⊂ Rational ⊂ Real
Think: Naughty Weasels Inside Red Rooms 🦦

★★☆☆☆ BASIC

Q9 of 20

Which set does the number −5 belong to? Select the most specific set.

💡 EXPLANATION

−5 is a negative integer. It belongs to integers, rationals, and real numbers — but the most specific category is negative integer.
Natural numbers = {1, 2, 3...} → −5 not included ❌
Whole numbers typically = {0, 1, 2...} → −5 not included ❌
Integers = {...−3, −2, −1, 0, 1, 2, 3...} → −5 ✅

★★★☆☆ MEDIUM

Q10 of 20

Which equation has a solution that is irrational?

💡 EXPLANATION

• 2x + 3 = 0 → x = −3/2 (rational) ✅
• x² + 1 = 0 → x² = −1 → no real solution! (imaginary)
• x + √4 = 0 → x = −√4 = −2 (rational, since √4=2) ✅
• x² = 2 → x = ±√2 = ±1.41421... → IRRATIONAL ✅

√2 cannot be written as a fraction — this is a famous proof from ancient Greece!

✦ SECTION 4 · Operations & Tricky Problems ✦

WATCH OUT Operations on rationals:
rational + rational = rational | rational × irrational = ??? (usually irrational)
Exception: 0 × √2 = 0 (rational!) 😲

★★★☆☆ MEDIUM

Q11 of 20

What is 34 + 56 in simplest form?

STEP

Find LCM of 4 and 6 first! LCM(4,6) = 12

💡 EXPLANATION

LCM(4, 6) = 12
3/4 = 9/12
5/6 = 10/12
9/12 + 10/12 = 19/12 = 1 and 7/12 ✅

⚠️ Common mistake: adding numerators AND denominators (3+5)/(4+6) = 8/10 ❌ WRONG!

★★★☆☆ MEDIUM

Q12 of 20

Simplify: 23 × 914

💡 EXPLANATION

Cross-cancel first (saves work!):
2 and 14 share factor 2: → 2/14 becomes 1/7
3 and 9 share factor 3: → 3/9 becomes 1/3... wait, flip it: 9/3 = 3
So: (2/3) × (9/14) = (2×9)/(3×14) = 18/42 = 3/7 ✅

Or cancel diagonally: ²/₃ × ⁹/₁₄ → cancel 2&14 to 1&7, cancel 3&9 to 1&3 → 3/7

★★★★☆ HARD

Q13 of 20

Tricky! Which of the following is always rational?

💡 EXPLANATION

• √2 + √3 = irrational (two different irrationals added) ❌
• √2 × √3 = √6 = irrational ❌
• √2 + √2 = 2√2 = irrational ❌
• √2 × √2 = 2 → RATIONAL! ✅

Key insight: √a × √a = a always! An irrational times itself = rational. This is the famous trick examiners love! 🎯

★★★★☆ HARD

Q14 of 20

Between which two consecutive integers does √50 lie?

THINK

Find perfect squares near 50: 7² = 49, 8² = 64...

💡 EXPLANATION

7² = 49 and 8² = 64
Since 49 < 50 < 64 → √49 < √50 < √64 → 7 < √50 < 8 ✅

Note: √50 = √(25×2) = 5√2 ≈ 7.07 — both D and B describe the same number, but B answers what was asked (between which integers).

★★★★☆ HARD

Q15 of 20

If the repeating decimal 0.36̄ (= 0.3666...) is written as a fraction ab in lowest terms, what is a + b?

💡 EXPLANATION

x = 0.3666...
10x = 3.666...
100x = 36.666...
100x − 10x = 36.666... − 3.666...
90x = 33 → x = 33/90 = 11/30
a + b = 11 + 30 = 41 ✅

⚠️ GCF(33,90) = 3, so simplify: 33÷3=11, 90÷3=30

✦ SECTION 5 · Number Line & Ordering ✦

NUMBER LINE Left = smaller, Right = larger
Negative numbers: −3 < −1 (further left = smaller value!) 🔢

★★☆☆☆ BASIC

Q16 of 20

Order from smallest to largest: −2, 0.5, −0.3, 1, −1.5

←

−2

−1

→

💡 EXPLANATION

On the number line, left = smaller:
−2 is furthest left → smallest
Then −1.5, then −0.3, then 0.5, then 1
Answer: −2 < −1.5 < −0.3 < 0.5 < 1 ✅

⚠️ Common mistake: thinking −2 > −0.3 because 2 > 0.3 — WRONG for negatives!

★★★☆☆ MEDIUM

Q17 of 20

Which statement about rational numbers is FALSE?

💡 EXPLANATION

• Every integer IS rational: e.g., 5 = 5/1 ✅ TRUE
• 0 IS rational: 0 = 0/1 ✅ TRUE
• NOT every rational is an integer: 1/2 is rational but NOT an integer ❌ FALSE!
• Terminating decimals ARE rational ✅ TRUE

Integers ⊂ Rationals, but Rationals ⊄ Integers. The arrow only goes one way!

★★★★☆ HARD

Q18 of 20

⚠️ Classic Trap! Is 0.999... (= 0.9̄) equal to 1?

HINT

Use the algebra trick: Let x = 0.999... → 10x = 9.999... → 10x − x = ?

💡 EXPLANATION

Let x = 0.999...
10x = 9.999...
10x − x = 9.999... − 0.999...
9x = 9 → x = 1

Therefore 0.999... = 1 exactly! ✅ This is mathematically proven, not an approximation. It surprises most students — that's why examiners love it! 🎯

★★★★★ CHALLENGE

Q19 of 20

How many rational numbers exist between 0 and 1 on the number line?

💡 EXPLANATION

Between ANY two rational numbers, you can always find another! For example:
Between 0 and 1: try 1/2
Between 0 and 1/2: try 1/4
Between 0 and 1/4: try 1/8... and so on forever!

This is called the density property of rational numbers. There are infinitely many! 🌌

★★★★★ BOSS LEVEL

Q20 of 20 — FINAL BOSS 🐉

A student says: "If p and q are both irrational, then p + q must also be irrational."
Is this statement always true?

THINK — Try a counterexample!

What if p = √2 and q = −√2? What is p + q?

💡 EXPLANATION

The statement is FALSE! Counterexample:
p = √2 (irrational) and q = −√2 (irrational)
p + q = √2 + (−√2) = 0 → rational! ✅

In math, to disprove "always true" you only need ONE counterexample.
This is a favourite exam trap — always look for counterexamples! 🏆

However: rational + irrational = ALWAYS irrational (that one IS always true).

Quiz Complete!

📐 Rational Numbers