Middle School Math Quiz: Place Value

7 questions · exam conditions

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Place ValueQuestion 1 of 7

A scientist measures the thickness of a cell membrane as $0.000000847$ meters. When this measurement is rounded to the nearest hundred-millionth, the result can be written in the form $a \times 10^{-7}$ where $a$ is a whole number. What is the value of $a$ ?

847

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Middle School Math Quiz

Middle School Math Quiz: Place Value

Practice Place Value in Middle School Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Place Value, giving you a quick way to practice the rules, question types, and explanations that matter most for Middle School Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

8 (correct answer)
85
9
847

Explanation: First, identify what place value represents hundred-millionths: that's the 8th decimal place (10⁻⁸). The number 0.000000847 has 8 in the hundred-millionths place, 4 in the billionths place, and 7 in the ten-billionths place. To round to the nearest hundred-millionth, look at the billionths place (4). Since 4 < 5, round down, keeping the hundred-millionths digit as 8 and dropping the rest. This gives 0.00000080 = 8 × 10⁻⁷. Choice B (85) would result from incorrectly including the next digit. Choice C (9) would result from incorrectly rounding up. Choice D (847) ignores the rounding requirement entirely.

Question 2

Consider the number $N = 567.89abc$ where $a$ , $b$ , and $c$ represent unknown digits. If $N$ rounded to the nearest hundredth equals $567.89$ exactly, and $N$ rounded to the nearest thousandth equals $567.891$ , what is the sum $a + b + c$ ?

12 (correct answer)
14
13
11

Explanation: For N to round to 567.89 when rounded to the nearest hundredth, the digit in the thousandths place (a) must be less than 5, so a ∈ {0,1,2,3,4}. For N to round to 567.891 when rounded to the nearest thousandth, we need a = 1 and the ten-thousandths digit (b) must be at least 5, so b ∈ {5,6,7,8,9}. Since N rounds to exactly 567.891 (not 567.892), and b ≥ 5 causes rounding up to 567.891, we need the hundred-thousandths digit (c) to be less than 5 to avoid further rounding up. For the sum to equal 12: a = 1, and b + c = 11. The only valid combination is b = 9, c = 2 (since c < 5). Check: 567.89192 rounds to 567.89 (hundredths) and 567.891 (thousandths). Sum: 1 + 9 + 2 = 12.

Question 3

A GPS device displays distances with varying precision. It shows the distance to a destination as $47.3$ miles. If the actual distance could be anywhere from $47.25$ to $47.34$ miles, and the GPS rounds to the nearest tenth, what is the greatest possible value for the digit in the thousandths place of the actual distance?

4
5
8
9 (correct answer)

Explanation: For a distance to round to 47.3 when rounded to the nearest tenth, it must be in the range [47.25, 47.35). Since we're told the actual distance is between 47.25 and 47.34 miles, we need to find the maximum possible digit in the thousandths place. The largest value in the given range is just under 47.35, which could be 47.349... The digit in the thousandths place of 47.349 is 9. Choice A (4) represents the minimum thousandths digit. Choice B (5) would be correct for the minimum case. Choice C (8) is plausible but not the maximum.

Question 4

A digital scale displays weights to the nearest gram. The scale shows three different readings throughout the day: 847 g in the morning, 851 g at noon, and 849 g in the evening.

If the actual weight at noon was $850.7$ grams, what is the range of possible actual weights (in grams) for the morning reading, assuming the scale's rounding behavior is consistent?

846.5 to 847.4
846.0 to 848.0
846.5 to 847.5 (correct answer)
847.0 to 847.9

Explanation: When you encounter rounding problems, you need to understand the range of values that round to a specific displayed number. Since this scale rounds to the nearest gram, any actual weight from 846.5 up to (but not including) 847.5 grams would display as 847 g. Let's verify this logic using the given information. The noon reading shows 851 g, and we know the actual weight was 850.7 g. Since 850.7 is closer to 851 than to 850, it correctly rounds to 851 g. This confirms the scale rounds to the nearest whole number. For any displayed value, the actual weight falls within a range of ±0.5 grams around whole numbers. Specifically, a reading of 847 g means the actual weight is anywhere from 846.5 g (inclusive) up to 847.5 g (exclusive). However, since we're dealing with continuous measurements, we typically express this as 846.5 to 847.5 grams. Looking at the wrong answers: Choice A (846.5 to 847.4) cuts off too early at 847.4, missing values like 847.45 that would still round to 847. Choice B (846.0 to 848.0) is far too wide—weights like 846.2 would round to 846, not 847, and 847.8 would round to 848. Choice D (847.0 to 847.9) misses the lower half of the range entirely, excluding values like 846.8 that clearly round up to 847. Remember: when a value rounds to the nearest whole number, the actual range extends 0.5 units below and 0.5 units above that displayed number.

Question 5

The decimal $0.4059$ is written in expanded form as $4 \times 10^{-1} + 0 \times 10^{-2} + 5 \times 10^{-3} + 9 \times 10^{-4}$ . What is the value of the digit in the ten-thousandths place when $0.4059$ is multiplied by $10^2$ ?

5
9 (correct answer)
0
4

Explanation: When 0.4059 is multiplied by 10², each digit shifts two places to the left. The result is 40.59. In this number, the ten-thousandths place contains the digit 9 (since 40.59 = 40.5900). Choice A (5) represents the value in the hundredths place. Choice C (0) would be correct if students forgot that 40.59 has implied zeros, making it 40.5900. Choice D (4) represents the tens place value.

Question 6

Three students compared the decimal numbers $2.4056$ , $2.405$ , and $2.4$ . Sarah claims that when all three numbers are rounded to the same number of decimal places, two of them become equal. What is the minimum number of decimal places needed to make Sarah's claim true?

1 decimal place
2 decimal places (correct answer)
3 decimal places
4 decimal places

Explanation: Testing different decimal places: At 1 decimal place: 2.4056→2.4, 2.405→2.4, 2.4→2.4 (all three equal, not just two). At 2 decimal places: 2.4056→2.41, 2.405→2.41, 2.4→2.40 (two equal: 2.4056 and 2.405 both round to 2.41). This satisfies Sarah's claim with the minimum number of decimal places. Choice A makes all three equal, not two. Choice C (3 places) works but isn't the minimum. Choice D (4 places) makes all numbers different.

Question 7

The number $4.0506$ can be written as $4 + 5 \times 10^{-2} + 6 \times 10^{-4}$ . A student claims that when this number is multiplied by $10^3$ and then rounded to the nearest ten, the result has exactly two non-zero digits. Is the student's claim correct, and what is the result?

Yes, the result is 4060
No, the result is 4051
Yes, the result is 4050 (correct answer)
No, the result is 4100

Explanation: When you see a number written in expanded form using powers of 10, you're working with place value concepts that connect to scientific notation and decimal operations. The key here is to carefully follow the order of operations: multiply first, then round. Let's work through this step by step. First, multiply

4.0506

10^3 = 1000

4.0506 \times 1000 = 4050.6

Next, round

4050.6

to the nearest ten. Since the ones digit is 0 (which is less than 5), we round down to

4050

. Now count the non-zero digits in

4050

: the digits 4 and 5 are non-zero, while the two zeros are not. That's exactly two non-zero digits, so the student's claim is correct. Looking at the wrong answers: Choice A gives

4060

, which would result from incorrectly rounding

4050.6

up instead of down—remember that you only round up when the digit being dropped is 5 or greater. Choice B gives

4051

, which represents the error of rounding to the nearest whole number instead of the nearest ten. Choice D gives

4100

, which involves multiple errors in both the multiplication and rounding steps. When working with place value and rounding problems, always complete operations in the correct order and pay close attention to what place you're rounding to. Double-check your work by counting digits carefully—zeros are still digits, but they're not non-zero digits.