Challenge Questions – Place Value

These challenge questions are designed to push your place value knowledge to its limits. They combine multiple topics, require logical reasoning, and demand precise application of concepts. Tackle them one at a time and show all working.

Challenge Set A – Reasoning with Place Value

1. A 6-digit number has the digit 8 in the ten-thousands place and the digit 3 in all other places. What is the number? Write it in words and expanded form.
2. The digit 5 in a certain number has a place value of 50,000. The digit 7 in the same number has a place value that is 10 times its face value. Write a possible 6-digit number satisfying both conditions.
3. I am a 7-digit number. My millions digit is the largest single digit. My ones digit is half my millions digit. The sum of all my digits is 25. All my other digits are zeros. What number am I?
4. A number rounded to the nearest thousand gives 7,000. Rounded to the nearest hundred it gives 6,900. What are the possible values of the original number?
5. Two numbers each have 5 digits. Their sum has 6 digits. What are the smallest possible values of each number?

Challenge Set B – Decimal and Measurement Problems

6. Three lengths are measured as 1.8 m, 1.75 m, and 1.805 m. Arrange them in descending order and find their total to 3 decimal places.
7. A number is 1,000 times greater than 0.047. What is it, and what is its place value structure?
8. If you write every whole number from 1 to 100, how many times does the digit 7 appear? (Hint: count both place values.)
9. A price is £3.49. After a 10% increase, what is the new price? Express both as decimals and state the place value of each digit in the new price.
10. The distance from Earth to the Moon is approximately 384,400 km. Write this in scientific notation. How many significant figures does the given value have?

Challenge Set C – Number Bases

11. Convert decimal 100 to binary, octal and hexadecimal. Show all working.
12. Add binary 1101 and 1011. Show binary working (with carrying) and verify by converting both to decimal first.
13. The hexadecimal colour code for a certain shade of blue is #0000FF. What decimal value does the blue channel have, and what is its significance as a power of 16?
14. A computer file is 65,536 bytes. Express this as a power of 2 and as a power of 16.
15. Which base-10 number is represented by the polynomial 3x² + 2x + 1 when (a) x = 10, (b) x = 2, (c) x = 16?

Challenge Set D – Scientific Notation and Sig Figs

16. The speed of light is 2.998 × 10⁸ m/s. How far does light travel in one hour? Express your answer in scientific notation to 4 significant figures.
17. A proton has a mass of 1.67 × 10⁻²⁷ kg. How many protons would be needed to have a total mass of 1 kg? Express your answer in scientific notation.
18. Round 0.0007654 to 2 sig figs, 3 sig figs, and 4 sig figs. Show each result.
19. Two measurements are 4.50 × 10³ and 3.2 × 10² kg. Find their sum in scientific notation.
20. A googol (10¹⁰⁰) is written as a decimal. How many digits does it have, and how many of them are zeros?

Challenge Set E – Multi-Step Problems

21. A 9-digit number contains the digits 1 through 9 each exactly once. The digit in the hundred millions place is 5 and the digit in the ones place is 3. The number is greater than 500,000,000. List three possible values.
22. Prove algebraically that any 3-digit number "abc" (hundreds = a, tens = b, ones = c) is equal to 100a + 10b + c. Then show that the 3-digit number formed by reversing its digits minus the original always gives a difference that is divisible by 99.
23. Using only the digits 3, 5, 7 and 0 (each used at most once), write: (a) the largest possible 4-digit number; (b) the smallest possible 4-digit number; (c) the number closest to 5,000.
24. If you convert a 4-digit decimal number to binary and the binary representation has 12 bits, what are the possible range of values for the original decimal number?
25. Design a "place value puzzle": create a number where the place value of the digit in the thousands place plus the face value of the digit in the hundreds place equals exactly 1,009. Find at least three solutions.

Summary

These 25 challenge questions require you to combine multiple place value concepts, reason carefully about digit positions, and apply mathematical rules precisely. They are excellent preparation for competitions, advanced examinations, and any situation that demands rigorous numerical thinking. Work through them systematically, and remember: showing clear working is as important as reaching the correct answer.