Math Formative Intermediate 1

Question 1

Order the following numbers from smallest to largest: -√50, -7.1, -7, -π, 0.

Answer 1

-√50 (~-7.071), -7.1, -7, -π (~-3.1416), 0

Question 2

Mention three irrational numbers greater than 5 and explain why they are irrational.

Answer 2

Examples: √29, π+5, e^2. All have nonterminating, nonrepeating decimals and cannot be expressed as a/b with integers.

Question 3

Compare √2 and 1.5. Which is greater and why?

Answer 3

√2 ≈ 1.4142 is less than 1.5, because 1.5 - √2 ≈ 0.0858 > 0.

Question 4

Which is larger: 3/7 or 0.43? Show detailed reasoning.

Answer 4

3/7 ≈ 0.428571 < 0.43, so 0.43 is larger.

Question 5

Write five rational numbers between 2/3 and 3/4.

Answer 5

Examples: 0.68, 0.69, 0.7, 11/16, 0.74.

Question 6

True or False: All square roots are irrational. Explain.

Answer 6

False. Perfect square roots (e.g., √25=5) are rational.

Question 7

Order from largest to smallest: π, 22/7, √10, 3.15.

Answer 7

√10 ≈ 3.1623, 22/7 ≈ 3.1429, π ≈ 3.1416, 3.15 — so order: √10, 3.15, 22/7, π.

Question 8

A diver descends 27 meters below sea level and then ascends 8 meters. What is the final position relative to sea level?

Answer 8

Start: -27, ascend 8 → -19 meters.

Question 9

Is 0.101001000100001… rational or irrational? Explain.

Answer 9

Irrational, because it has a nonterminating, nonrepeating decimal pattern.

Question 10

Write three numbers between √2 and √3.

Answer 10

Examples: 1.42, 1.45, 1.7.

Question 11

Compare −4.25 and −17/4. Explain.

Answer 11

−4.25 = −17/4 exactly; they are equal.

Question 12

Which is closer to zero: -0.005 or 1/(-400)?

Answer 12

1/(-400) = -0.0025, which is closer to zero.

Question 13

Give two examples of rational numbers that are repeating decimals.

Answer 13

Examples: 1/3 = 0.333…, 5/6 = 0.8333…

Question 14

Is the product of a nonzero rational number and an irrational number always irrational? Explain.

Answer 14

Yes, because multiplying by a nonzero rational scales the irrational but does not make it rational.

Question 15

Compare the cube root of 125 and √50.

Answer 15

∛125=5, √50≈7.071, so √50 is greater.

Question 16

Which is larger: 4/√2 or √8/2?

Answer 16

Both equal √8/2 ≈ 1.4142, so they are equal.

Question 17

Order from smallest to largest: 1.01, 101/100, 1009/1000.

Answer 17

All are equal to 1.01, so they are the same.

Question 18

Is -√(49/4) rational?

Answer 18

Yes, -√(49/4) = -7/2 = -3.5.

Question 19

A hiker is at an altitude of 1200 m, then descends 1350 m. What is the final altitude?

Answer 19

1200 - 1350 = -150 m (150 m below sea level).

Question 20

Compare √5 and 2.3 using < or >.

Answer 20

√5 ≈ 2.236 < 2.3.

Question 21

Write two distinct irrational numbers between 7 and 8.

Answer 21

Examples: √53, π+4.

Question 22

True or False: Every nonterminating decimal is irrational.

Answer 22

False. Some nonterminating decimals are repeating and rational (e.g., 0.333…).

Question 23

Is the sum of two irrational numbers always irrational? Give a counterexample if not.

Answer 23

No. Example: √2 + (2 - √2) = 2, which is rational.

Question 24

Compare 1/(√2) and √2/2.

Answer 24

They are equal; both ≈ 0.7071.

Question 25

Write four rational numbers between -1/2 and 0.

Answer 25

Examples: -0.49, -0.3, -1/4, -0.01.

Question 26

Order from smallest to largest: -π, -3.14, -3 1/7.

Answer 26

-3.14, -π ≈ -3.1416, -3 1/7 ≈ -3.142857 — so order: -3 1/7, -π, -3.14.

Question 27

Is √(2/3) rational?

Answer 27

No, because both numerator and denominator produce an irrational when square-rooted.

Question 28

Write three different fractions equal to 0.4.

Answer 28

2/5, 4/10, 40/100.

Question 29

Which is larger: cube root of 2 or 2/3?

Answer 29

Cube root of 2 ≈ 1.26 > 2/3 ≈ 0.666...

Question 30

A bank balance goes from -$500 to -$200. Did it increase or decrease? By how much?

Answer 30

Increased by $300.

Question 31

Compare 1.414 and √2 with reasoning.

Answer 31

√2 ≈ 1.414213..., so 1.414 < √2.

Question 32

Give two rational approximations of π accurate to 3 decimal places.

Answer 32

3.142, 355/113 ≈ 3.141593.

Question 33

Write two irrational numbers between -4 and -3.

Answer 33

Examples: -√10, -π.

Question 34

Is -7 an integer, a rational, and a real number?

Answer 34

Yes, -7 is all three.

Question 35

Order from largest to smallest: 5/√2, √8, 2√2.

Answer 35

√8=2√2 ≈ 2.8284, 5/√2 ≈ 3.5355, so order: 5/√2, √8, 2√2.

Question 36

Compare √(49) and √(50).

Answer 36

√49=7, √50≈7.071, so √50 is greater.

Question 37

Write five rational numbers between -2 and -1.

Answer 37

Examples: -1.9, -1.75, -3/2, -1.25, -1.01.

Question 38

Is √0.04 rational?

Answer 38

Yes, √0.04=0.2, which is rational.

Question 39

True or False: The reciprocal of an irrational number is irrational.

Answer 39

True, except if the irrational number is 0 (undefined reciprocal).

Question 40

Which is larger: 0.666... or 2/3?

Answer 40

They are equal; 0.666... = 2/3.

Question 41

Order from smallest to largest: -1/3, -0.3, -0.333...

Answer 41

-0.333..., -1/3, -0.3 (note: -1/3 ≈ -0.3333...).

Question 42

Compare the magnitude of -8 and 7.

Answer 42

|-8|=8 > |7|=7, so -8 has greater magnitude.

Question 43

Write a number that is both a perfect square and a perfect cube.

Answer 43

64 (8^2=64 and 4^3=64).

Question 44

Is √7+√7 rational?

Answer 44

√7+√7=2√7, irrational.

Question 45

Give an example of a rational number between √5 and √6.

Answer 45

Example: 2.3 (since √5≈2.236, √6≈2.449).

Question 46

True or False: An integer can be irrational.

Answer 46

False. All integers are rational.

Question 47

Compare π^2 and 10.

Answer 47

π^2≈9.8696 < 10.

Question 48

Write the decimal form of 7/12 to 4 decimal places.

Answer 48

7/12 ≈ 0.5833.