Math worksheet for comparing real numbers, focusing on rational approximations and estimation of irrational numbers.

Let’s solve each problem one by one, step by step. We’ll compare real numbers — some are rational (like fractions or decimals), and some are irrational (like π or square roots). To compare them, we’ll use decimal approximations where needed.

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Problem 1: Which expression has the smallest value?

Options:
1) –π ≈ –3.14
2) –√10 ≈ –3.16 (since √9 = 3, √16 = 4 → √10 ≈ 3.16)
3) –16/5 = –3.2
4) –3.02

Now list them as decimals:
- –3.14
- –3.16
- –3.2
- –3.02

Smallest means most negative → –3.2 is smallest

✔ Answer: 3) –16/5

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Problem 2: Which number has the greatest value?

Options:
1) 1 2/3 = 1.666...
2) √2 ≈ 1.414
3) π/2 ≈ 3.1416 / 2 ≈ 1.5708
4) 1.5

Compare:
- 1.666...
- 1.414
- 1.5708
- 1.5

Greatest is 1.666...

✔ Answer: 1) 1 2/3

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Problem 3: In which list are the numbers in order from least to greatest?

We need to approximate:

√3 ≈ 1.732
π ≈ 3.1416
3 1/3 = 3.333...
3.2 = 3.2

So actual values:
√3 ≈ 1.732
π ≈ 3.1416
3.2 = 3.2
3 1/3 ≈ 3.333

Order from least to greatest:
√3 < π < 3.2 < 3 1/3

Check options:

1) 3.2, π, 3 1/3, √3 → NO (starts with big number)
2) √3, 3.2, π, 3 1/3 → NO (3.2 > π, so wrong order)
3) √3, π, 3.2, 3 1/3 → YES! Matches our order
4) 3.2, 3 1/3, √3, π → NO

✔ Answer: 3) √3, π, 3.2, 3 1/3

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Problem 4: Which numbers are arranged from smallest to largest?

Approximate:

√9.1 → √9 = 3, √10≈3.16 → √9.1 ≈ 3.016
π ≈ 3.1416
3.14 = 3.14
22/7 ≈ 3.142857...

So:

√9.1 ≈ 3.016
3.14 = 3.14
π ≈ 3.1416
22/7 ≈ 3.142857

Order: √9.1 < 3.14 < π < 22/7

Check options:

1) 3.14, 22/7, π, √9.1 → NO
2) √9.1, π, 3.14, 22/7 → NO (π > 3.14, so should be after)
3) √9.1, 3.14, 22/7, π → NO (22/7 > π, so π should come before 22/7? Wait no — 22/7 ≈ 3.142857, π ≈ 3.14159 → so π < 22/7)

Wait — correction:

Actually:
√9.1 ≈ 3.016
3.14 = 3.14000
π ≈ 3.14159
22/7 ≈ 3.14286

So correct order: √9.1 < 3.14 < π < 22/7

Look at option 4:
4) √9.1, 3.14, π, 22/7 → YES!

Option 3 says: √9.1, 3.14, 22/7, π → that would mean 22/7 < π, but it’s not — 22/7 is bigger.

So only option 4 matches.

✔ Answer: 4) √9.1, 3.14, π, 22/7

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Problem 5: Which list is in order from smallest to largest?

Numbers: √10, 22/7, π, 3.1

Approximate:

√10 ≈ 3.162
22/7 ≈ 3.142857
π ≈ 3.14159
3.1 = 3.1

So:

3.1 < π < 22/7 < √10

Check options:

1) √10, 22/7, π, 3.1 → decreasing → NO
2) 3.1, 22/7, π, √10 → 22/7 > π, so this is wrong order between those two
3) π, 22/7, 3.1, √10 → starts with π, but 3.1 is smaller → NO
4) 3.1, π, 22/7, √10 → YES! Matches: 3.1 < π < 22/7 < √10

✔ Answer: 4) 3.1, π, 22/7, √10

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Problem 6: Which list shows the numbers |–0.12|, √(1/82), 1/8, 1/9 in order from smallest to largest?

First simplify:

|–0.12| = 0.12
√(1/82) = 1/√82 → √81=9, √100=10 → √82≈9.055 → so 1/9.055 ≈ 0.1104
1/8 = 0.125
1/9 ≈ 0.1111

So let’s write all as decimals:

√(1/82) ≈ 0.1104
1/9 ≈ 0.1111
|–0.12| = 0.12
1/8 = 0.125

Order: √(1/82) < 1/9 < |–0.12| < 1/8

Check options:

1) |–0.12|, 1/8, 1/9, √(1/82) → NO
2) 1/8, 1/9, √(1/82), |–0.12| → NO
3) √(1/82), |–0.12|, 1/9, 1/8 → NO ( |–0.12| = 0.12, 1/9≈0.111 → so 1/9 should come before |–0.12| )
4) √(1/82), 1/9, |–0.12|, 1/8 → YES! Matches our order

✔ Answer: 4) √(1/82), 1/9, |–0.12|, 1/8

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Problem 7: In which group are the numbers arranged in order from smallest to largest?

Numbers: π, 3.14, √9.86, 22/7

Approximate:

π ≈ 3.14159
3.14 = 3.14000
√9.86 → √9 = 3, √10≈3.162 → √9.86 ≈ ? Let's compute: 3.14² = 9.8596 → so √9.86 ≈ 3.14006 (very close to 3.14)
22/7 ≈ 3.142857

So:

3.14 = 3.14000
√9.86 ≈ 3.14006
π ≈ 3.14159
22/7 ≈ 3.142857

Order: 3.14 < √9.86 < π < 22/7

Check options:

1) π, 3.14, √9.86, 22/7 → NO
2) √9.86, 22/7, 3.14, π → NO
3) 22/7, 3.14, π, √9.86 → NO
4) 3.14, √9.86, π, 22/7 → YES!

✔ Answer: 4) 3.14, √9.86, π, 22/7

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Problem 8: Correct arrangement from smallest to greatest?

Terms: 3√2, 4 1/8, |–4.24|, ∛75

Compute each:

3√2 ≈ 3 × 1.414 ≈ 4.242
4 1/8 = 4.125
|–4.24| = 4.24
∛75 → ∛64=4, ∛125=5 → try 4.2³ = 4.2×4.2=17.64 ×4.2≈74.088 → close to 75 → 4.21³ = ? 4.21×4.21≈17.7241 ×4.21≈74.6 → 4.22³≈4.22×4.22=17.8084×4.22≈75.15 → so ∛75 ≈ 4.217

So:

4 1/8 = 4.125
|–4.24| = 4.24
3√2 ≈ 4.242
∛75 ≈ 4.217

Wait — let’s reorder numerically:

4.125 (4 1/8)
then ∛75 ≈ 4.217
then |–4.24| = 4.24
then 3√2 ≈ 4.242

So order: 4 1/8 < ∛75 < |–4.24| < 3√2

Check options:

1) 3√2, 4 1/8, |–4.24|, ∛75 → NO
2) ∛75, |–4.24|, 4 1/8, 3√2 → NO (4 1/8 is smallest)
3) 4 1/8, ∛75, |–4.24|, 3√2 → YES!
4) 4 1/8, |–4.24|, ∛75, 3√2 → NO (because ∛75 ≈4.217 < 4.24)

✔ Answer: 3) 4 1/8, ∛75, |–4.24|, 3√2

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Problem 9: Which inequality is true if x = 3.04 / 1.48, y = 1.99 + 0.33, z = (1.3)^3?

Compute each:

x = 3.04 ÷ 1.48
Let’s divide: 1.48 × 2 = 2.96 → 3.04 - 2.96 = 0.08 → so 2 + 0.08/1.48 ≈ 2 + 0.054 ≈ 2.054

y = 1.99 + 0.33 = 2.32

z = (1.3)^3 = 1.3 × 1.3 = 1.69; 1.69 × 1.3 = 2.197 → 2.197

So:

x ≈ 2.054
z ≈ 2.197
y = 2.32

Order: x < z < y

Check options:

1) y < z < x → NO
2) y < x < z → NO
3) x < z < y → YES
4) x < y < z → NO (y=2.32 > z=2.197)

✔ Answer: 3) x < z < y

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Final Answer:
1) 3
2) 1
3) 3
4) 4
5) 4
6) 4
7) 4
8) 3
9) 3