Free Math Worksheets

- Compare the number of zeros in the divisor with the change in place value. - Division by a number greater than \(1\) makes this positive number smaller.

1. \(15.4\div10=1.54\). Each digit has a value one tenth as large. 2. \(15.4\div1000=0.0154\). Each digit has a value one thousandth as large. 3. In general, dividing by \(10^n\) shifts every digit \(n\) place values to the right relative to the decimal point. Zeros are used as placeholders when needed.

\(15.4\div10=1.54\) and \(15.4\div1000=0.0154\). Dividing by a power of \(10\) shifts the digits to smaller place values according to the power of \(10\).

- How many places must the digits shift for the decimal to become a whole number? - What happens to place values when you multiply by \(10\), \(100\), or \(1000\)? - How is the exponent in a power of ten related to the number of place-value shifts?

1. For \(0.8\), multiplying by \(10^1\) moves each digit one place to the left: \(0.8 \times 10 = 8\). 2. For \(0.045\), multiplying by \(10^3\) moves each digit three places to the left: \(0.045 \times 1000 = 45\). 3. For \(1.203\), multiplying by \(10^3\) moves each digit three places to the left: \(1.203 \times 1000 = 1203\).

a) \(10^1\); product: \(8\) b) \(10^3\); product: \(45\) c) \(10^3\); product: \(1203\)

- Does a zero at the far right of a decimal change its value? - Rewrite the decimal without any unnecessary trailing zero. - Test powers of ten in order, starting with \(10^1\).

1. The decimals \(0.750\) and \(0.75\) have the same value because a trailing zero does not change a decimal's value. 2. Multiplying by \(10^1\) gives \(0.75 \times 10 = 7.5\), which is not a whole number. 3. Multiplying by \(10^2\) gives \(0.75 \times 100 = 75\), which is a whole number. 4. Therefore, \(10^2\), not \(10^3\), is the smallest power of ten that works.

The claim is incorrect. Since \(0.750 = 0.75\), the smallest power of ten is \(10^2\), because \(0.75 \times 100 = 75\).

- Track how the place value of the nonzero digits changes. - Multiplication by a power of \(10\) moves digits to greater place values. - Division by a power of \(10\) moves digits to smaller place values. - The exponent or number of zeros tells you how many place values change.

1. For a), \(0.082\times1000=82\). 2. For b), \(740\div1000=0.74\). 3. For c), \(0.005\times100{,}000=500\). 4. For d), \(1.23\div100=0.0123\).

a) \(1000\) b) \(1000\) c) \(100{,}000\) d) \(0.0123\)

- Compare the place value of the nonzero digits before and after division. - When a digit shifts to a smaller place value, zeros may be needed as placeholders.

1. For a), dividing \(68\) by \(100\) shifts the digits two place values to the right relative to the decimal point: \(68\div100=0.68\). 2. For b), \(0.9\div100=0.009\), so the missing divisor is also \(100\). 3. The zeros are placeholders that show the tenths and hundredths places before the \(9\) in the thousandths place.

a) \(100\) b) \(100\). The zeros are place-value placeholders.

- Think about the effect of dividing by \(1000\) and multiplying by \(100\) separately. - Compare the number of place-value shifts in opposite directions. - Combine the two changes into one power-of-ten operation.

1. Dividing by \(1000\) makes each digit move three place values to the right relative to the decimal point. 2. Multiplying by \(100\) moves each digit two place values back to the left. 3. The net effect is one place value to the right, which makes the number one tenth as large. 4. A single division by \(10\) has the same effect.

The overall effect is a division by \(10\). Each digit ends one place value to the right relative to the decimal point.

- Find the smallest power of ten for each decimal separately. - Begin with \(10^1\) and test larger powers only as needed. - Compare the exponents after you have found all four powers.

1. For \(A = 0.002\), the smallest power is \(10^3\), because \(0.002 \times 1000 = 2\). 2. For \(B = 0.05\), the smallest power is \(10^2\), because \(0.05 \times 100 = 5\). 3. For \(C = 0.8\), the smallest power is \(10^1\), because \(0.8 \times 10 = 8\). 4. For \(D = 1.23\), the smallest power is \(10^2\), because \(1.23 \times 100 = 123\). 5. Comparing the exponents gives \(C\), then \(B\) and \(D\), then \(A\).

\(C\) requires \(10^1\); \(B\) and \(D\) each require \(10^2\); \(A\) requires \(10^3\). Therefore, the order is \(C\), then \(B\) and \(D\), then \(A\).

- Track how the place value of each digit changes. - Think about what happens when a positive number is divided by a number less than \(1\). - Rewrite a decimal divisor using an equivalent power-of-ten relationship when useful.

1. For a), \(3.4\div100=0.034\). Each digit shifts two place values to the right relative to the decimal point. 2. For b), \(0.5\div100=0.005\), so the divisor is \(100\). 3. For c), \(0.002\times10{,}000=20\), so the multiplier is \(10{,}000\). 4. For d), \(0.8\div0.01=80\) and \(0.8\times100=80\). Dividing by \(0.01\) has the same effect as multiplying by \(100\).

a) \(0.034\); each digit shifts two place values to the right relative to the decimal point. b) \(100\) c) \(10{,}000\) d) Both equal \(80\). Dividing by \(0.01\) is equivalent to multiplying by \(100\).

- Start with the divisions by \(10\) and \(1\). - Look for a pattern in the quotients. - Track how the divisor changes from one line to the next. - Compare the place value of the digits in consecutive quotients.

1. The quotients are \(0.045\), \(0.45\), \(4.5\), \(45\), and \(450\). 2. Each time the divisor becomes one tenth as large, the quotient becomes ten times as large.

a) \(0.045\) b) \(0.45\) c) \(4.5\) d) \(45\) e) \(450\) The quotient becomes \(10\) times as large each time the divisor becomes one tenth as large.