Interpret Place Value - Praxis Core Math

1. How many hundredths are equivalent to the number 2.7?
   Answer: 270 The number 2.7 can be written as the fraction $\frac{27}{10}$. To find the equivalent number of hundredths, we need to find an equivalent fraction with a denominator of 100. $\frac{27}{10} = \frac{27 \times 10}{10 \times 10} = \frac{270}{100}$. This means 2.7 is equivalent to 270 hundredths.
2. A computer performs a calculation in 0.00000035 seconds. What is the place value of the digit 3 in this number?
   Answer: Ten-millionths To identify the place value, count the places to the right of the decimal point. The first place is tenths, then hundredths, thousandths, ten-thousandths, hundred-thousandths, millionths, and the seventh place is ten-millionths. The digit 3 is in the seventh decimal place, which is the ten-millionths place.
3. The number 0.47 is multiplied by 200. In the original number, the value of the digit 4 is 0.4. What is the value of the digit 4 in the product?
   Answer: 4 First, find the product: $0.47 \times 200 = 94$. In the resulting number, 94, the digit 4 is in the ones place. Therefore, its value is 4.
4. When a whole number is rounded to the nearest thousand, the result is 14,000. What is the smallest possible value for the original whole number?
   Answer: 13,500 To round to the nearest thousand, we look at the digit in the hundreds place. If the hundreds digit is 5 or greater, we round up. If it is 4 or less, we round down. The smallest whole number that rounds up to 14,000 is 13,500. Any number from 13,500 to 14,499 will round to 14,000.
5. A number is expressed in expanded form as $7 \times 10^3 + 4 \times 10^1 + 6 \times 10^{-2} + 1 \times 10^{-3}$. What is this number written in standard decimal form?
   Answer: 7,040.061 Evaluate each term based on its power of 10: $7 \times 10^3 = 7000$; $4 \times 10^1 = 40$; $6 \times 10^{-2} = 0.06$; $1 \times 10^{-3} = 0.001$. Summing these values gives $7000 + 40 + 0 + 0.06 + 0.001 = 7040.061$.
6. In the number 6,194,382.57, one digit has a value of 90,000 and another digit has a value of 0.07. What is the sum of these two digits?
   Answer: 16 First, identify the digits from their values. The digit with a value of 90,000 is 9 (in the ten thousands place). The digit with a value of 0.07 is 7 (in the hundredths place). The question asks for the sum of these two digits, which is $9 + 7 = 16$.
7. Using the digits 1, 0, 8, 5 without repetition, what is the smallest possible number greater than 1 that can be formed, using three of the digits to the right of the decimal point?
   Answer: 1.058 The number must be greater than 1, so the digit in the ones place must be at least 1. To make the number as small as possible, the ones digit should be the smallest available digit that is not zero, which is 1. To make the decimal part as small as possible, the remaining digits (0, 5, 8) should be arranged in increasing order to the right of the decimal. This gives .058. Combining the parts, the smallest number is 1.058.
8. The numbers $X=0.505$, $Y=0.55$, and $Z=0.5$ are to be placed in order from least to greatest. Which of the following is the correct order?
   Answer: Z, X, Y To compare decimals, it is helpful to write them with the same number of decimal places. $X = 0.505$, $Y = 0.550$, and $Z = 0.500$. Now, comparing the numbers as if they were whole numbers (505, 550, 500), the order from least to greatest is 500, 505, 550. This corresponds to the order Z, X, Y.
9. A cashier mistakenly enters $4.50 for an item that costs $45.00. The value the cashier entered is what fraction of the correct value?
   Answer: $\frac{1}{10}$ To find the fraction, divide the entered value by the correct value: $\frac{4.50}{45.00}$. This simplifies to $\frac{4.5}{45}$. Since $4.5 \times 10 = 45$, the fraction is equivalent to $\frac{1}{10}$. The error was a shift of one place value, which corresponds to a factor of 10.
10. A number has the form ABC.DE, where A, B, C, D, and E are non-zero digits. The value of the digit A is 1,000 times the value of the digit D. Which of the following statements must be true?
    Answer: The digit A is equal to the digit D. The digit A is in the hundreds place, so its value is $A \times 100$. The digit D is in the tenths place, so its value is $D \times \frac{1}{10}$. The problem states that $A \times 100 = 1000 \times (D \times \frac{1}{10})$. Simplifying the right side gives $A \times 100 = 100 \times D$. Dividing both sides by 100 yields $A = D$.
11. The number 9,412.87 is changed by decreasing the value of its tens digit by 20 and increasing the value of its hundredths digit by 0.05. What is the new number?
    Answer: 9,392.92 The original number is 9,412.87. The tens digit is 1, with a value of 10. Decreasing this value by 20 results in a new value of $10-20 = -10$. This is equivalent to decreasing the hundreds digit by 1 and making the tens digit 9. So 941_ becomes 939_. The hundredths digit is 7, with a value of 0.07. Increasing this by 0.05 gives $0.07 + 0.05 = 0.12$. This means the new hundredths digit is 2 and we carry 0.1 to the tenths place. The tenths digit becomes $8+1=9$. Combining changes: the number becomes 9,392.92. Alternatively, subtract 20 and add 0.05: $9412.87 - 20 + 0.05 = 9392.87 + 0.05 = 9392.92$.
12. A city's budget is approximately $12,755,000. How would the value of the digit 2 be read in words?
    Answer: Two million dollars In the number 12,755,000, the digit 2 is in the millions place. Therefore, its value is 2,000,000, which is read as 'two million'.
13. In which of the following numbers does the digit 8 represent a value that is $\frac{1}{100}$ of the value represented by the digit 8 in the number 18,234?
    Answer: 3,482 In the number 18,234, the digit 8 is in the thousands place, so its value is 8,000. We need to find a number where the value of the digit 8 is $\frac{1}{100} \times 8,000 = 80$. A digit 8 has a value of 80 when it is in the tens place. In the number 3,482, the digit 8 is in the tens place.
14. A number with two decimal places is rounded to the nearest tenth, giving 6.5. In the original number, the digit in the hundredths place is an odd number that is also a factor of the digit in the tenths place. What was the original number?
    Answer: 6.51 If a number rounds to 6.5, it must be in the range [6.45, 6.55). Let the number be 6.AB. If A=4, the number is 6.4B. It rounds to 6.5 only if B ≥ 5. The odd digits for B are 5, 7, 9. None of these are factors of A=4. So, the tenths digit cannot be 4. If A=5, the number is 6.5B. It rounds to 6.5 only if B < 5. The odd digits for B are 1, 3. We check if these are factors of A=5. 1 is a factor of 5. 3 is not a factor of 5. Therefore, the hundredths digit B must be 1. The original number is 6.51.
15. Let N be the number 843.15. A new number, M, is created by swapping the digit in the tens place of N with the digit in the tenths place of N. What is the positive difference between N and M?
    Answer: 29.70 In the number N=843.15, the tens digit is 4 and the tenths digit is 1. Swapping them creates the number M=813.45. The difference is $N - M = 843.15 - 813.45 = 29.70$.
16. The value of the digit 7 in the number 471.3 is 70. If this number is divided by 1,000, what is the value of the digit 7 in the new, resulting number?
    Answer: 0.07 First, perform the division: $471.3 \div 1,000 = 0.4713$. In the new number, 0.4713, the digit 7 is in the hundredths place. Therefore, its value is $7 \times \frac{1}{100}$, which is 0.07.
17. Let K be the largest 4-digit number that can be formed using the digits 3, 1, 9, 6, each used once. Let L be the smallest 4-digit number that can be formed using the same digits. What is the value of K - L?
    Answer: 8,262 To form the largest number K, arrange the digits in descending order: 9,631. To form the smallest number L, arrange the digits in ascending order: 1,369. The difference is $K - L = 9631 - 1369 = 8262$.
18. For the number 2,781.9, what is the sum of the value of the digit in the hundreds place and the value of the digit in the tenths place?
    Answer: 700.9 The question asks for the sum of the values of the digits, not the sum of the digits themselves. The digit in the hundreds place is 7, and its value is 700. The digit in the tenths place is 9, and its value is 0.9. The sum of these values is $700 + 0.9 = 700.9$.
19. Which of the following expressions is equivalent to 3 ones and 14 tenths?
    Answer: 4.4 This question requires regrouping place values. '14 tenths' is equal to $\frac{14}{10}$, which is 1.4. This can be thought of as 1 one and 4 tenths. Adding this to the original 3 ones gives a total of $3 + 1 = 4$ ones and 4 tenths. The number is 4.4.
20. In the number 74,712, the value of the digit 7 in the ten thousands place is how many times the value of the digit 7 in the hundreds place?
    Answer: 100 The digit 7 in the ten thousands place has a value of 70,000. The digit 7 in the hundreds place has a value of 700. To find how many times greater the first value is than the second, divide them: $70,000 \div 700 = 100$. Therefore, the value is 100 times greater.

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