A rational number in its lowest form has denominator 23 × 5. How many decimal places will its decimal expansion have? Explain your answer.
Answer: 3 decimal places. p23 × 5 = 25p103, and because p is odd and not a multiple of 5 (lowest form), 25p ends in 5, so the third decimal digit is non-zero.
Step-by-step solution
Idea: Make the denominator a power of 10 by supplying the missing 5s: the highest power among 23 and 51 is 3, so we need 103. Then check the last digit is not 0.
- Let the number be p23 × 5 in lowest form. Multiply top and bottom by 52: 25p23 × 53 = 25p1000, so the decimal has at most 3 places.1 mark
- Lowest form means p has no factor 2 or 5, so p is odd. Then 25p is an odd multiple of 25, which ends in 25 or 75; its last digit is 5, not 0. So the third decimal place is non-zero: exactly 3 decimal places. (E.g. 140 = 0.025, 340 = 0.075.)1 mark
Exactly 3 decimal places, since p/(2³ × 5) = 25p/10³ and 25p ends in 5.
Check: 740 = 0.175, 1140 = 0.275, 3940 = 0.975: each has 3 places ✓.
Answer to write in the exam
p23 × 5 = p × 5223 × 53 = 25p103
Lowest form ⇒ p odd, 5 ∤ p ⇒ 25p ends in 5
∴ Exactly 3 decimal places (e.g. 340 = 0.075)
Common mistakes that cost marks
- Adding the powers (3 + 1 = 4) and answering 4 places. The number of places is the larger power, 3.
- Answering 1 place by looking at the single 5.
- Not explaining why the last digit cannot be 0 (it needs the lowest-form condition).
How this can come in the exam
MCQ (1 mark)
A rational number in lowest form has denominator 22 × 54. Its decimal expansion terminates after
- 2 places
- 4 places
- 6 places
- 8 places
Show answer
(B) 4 places
Multiply by 22 to get 104: 4 places.
Short answer (2 marks)
Write 980 as a decimal by making the denominator a power of 10. How many decimal places does it have?
Show answer
80 = 24 × 5, so multiply by 53 = 125: 112510000 (1 mark) = 0.1125, 4 decimal places (1 mark).Try one yourself
How many decimal places does a lowest-form rational number with denominator 2 × 53 have?
Show answer
3 places: p250 = 4p1000, and 4p does not end in 0 since 5 ∤ p.
More questions like this
- Let a = 712 and b = 56. Express both a and b in the form k1m and k2m where k1, k2 and m are integers and k2 − k1 > 6. Using the same denominator m, write exactly five distinct rational numbers lying between a and b keeping an integer numerator. Explain why the condition k2 − k1 > n + 1 is necessary to find n such rational numbers between the two rational numbers a and b using this method.
- Three rational numbers x, y, z satisfy x + y + z = 0 and xy + yz + zx = 0. Show that all the rational numbers x, y, z must be simultaneously zero.
- Show that the rational number (a + b)2 lies between the rational numbers a and b.
- Find the lengths of the hypotenuses of all the right triangles in the figure which is referred to as the square root spiral.
- Imagine you are living thousands of years ago in a small agricultural settlement along the banks of the Saraswati river. You have a herd of cattle. Every morning, they go out into the dense forests to graze, and every evening, they return. How do you ensure that a calf has not wandered off?