Practice · Decimal point position predicts the product · Sensim Math

Variant 1 easy

How many times as large as $ⓑ$ is $ⓐ$ ?

$ⓐ$ $21 \times 14$
$ⓑ$ $0.21 \times 0.14$

Your answer

Show solution

1 · Understandwhat's really being asked

Two products use the same digits. The first multiplies 21 by 14; the second multiplies 0.21 by 0.14, the same digits with the decimal point moved. We must say how many times bigger the first is.

Givens

The first: 21 times 14.
The second: 0.21 times 0.14.
Both use the same digits; only the decimal points differ.

Unknowns

How many times as large the first product is as the second.

Constraints

Neither product needs to be worked out to compare them.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem

Compare the two multiplications factor by factor. Each factor is smaller by a power of ten, and the effects combine, so the answer follows from the shifts alone.

3 · Execute3 carry out the plan

1Compare each factor

#5 Look for a Pattern 5.NBT.A.2

0.21 is 21 with the point moved 2 places, so it is 100 times smaller. Likewise 0.14 is 100 times smaller than 14.

0.21 = 21 \div 100,\quad 0.14 = 14 \div 100

Each factor shrank by its own power of ten.

2See how the product changes

#9 Solve an Easier Related Problem 5.NBT.A.2

Shrinking one factor 100 times and the other 100 times shrinks the product by both together.

100 \times 100 = 10000

The decimal point in the answer moves 4 places, so the first product is 10000 times the second.

3Check by direct calculation

#5 Look for a Pattern 5.NBT.B.7

Working both out confirms it: the digits are identical and only the point has moved 4 places.

21 \times 14 = 294,\quad 0.21 \times 0.14 = 0.0294

294 is 10000 times 0.0294.

Answer: 10000 times

4 · Reviewdoes it hold up?

Count the decimal places: the second product's factors have 2 and 2, so its product has 4 -- exactly the number of places the point moved, which is why the ratio is 10000.

Another way: Divide the two products directly: 294 divided by 0.0294 is 10000. Slower, and it hides the reason.

Standardsmin grade 5

5.NBT.A.2 Explain patterns in number of zeros and placement of decimal point — Reading each decimal shift as a division by a power of ten, and combining them.
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Confirming the two products directly.

💡Takeaway. Moving the point in each factor moves it in the answer too, and the moves add up -- so you can tell the ratio without multiplying.

Variant 2 easy

How many times as large as $ⓑ$ is $ⓐ$ ?

$ⓐ$ $36 \times 25$
$ⓑ$ $3.6 \times 0.25$

Your answer

Show solution

1 · Understandwhat's really being asked

Two products use the same digits. The first multiplies 36 by 25; the second multiplies 3.6 by 0.25, the same digits with the decimal point moved. We must say how many times bigger the first is.

Givens

The first: 36 times 25.
The second: 3.6 times 0.25.
Both use the same digits; only the decimal points differ.

Unknowns

How many times as large the first product is as the second.

Constraints

Neither product needs to be worked out to compare them.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem

Compare the two multiplications factor by factor. Each factor is smaller by a power of ten, and the effects combine, so the answer follows from the shifts alone.

3 · Execute3 carry out the plan

1Compare each factor

#5 Look for a Pattern 5.NBT.A.2

3.6 is 36 with the point moved 1 place, so it is 10 times smaller. Likewise 0.25 is 100 times smaller than 25.

3.6 = 36 \div 10,\quad 0.25 = 25 \div 100

Each factor shrank by its own power of ten.

2See how the product changes

#9 Solve an Easier Related Problem 5.NBT.A.2

Shrinking one factor 10 times and the other 100 times shrinks the product by both together.

10 \times 100 = 1000

The decimal point in the answer moves 3 places, so the first product is 1000 times the second.

3Check by direct calculation

#5 Look for a Pattern 5.NBT.B.7

Working both out confirms it: the digits are identical and only the point has moved 3 places.

36 \times 25 = 900,\quad 3.6 \times 0.25 = 0.900

900 is 1000 times 0.900.

Answer: 1000 times

4 · Reviewdoes it hold up?

Count the decimal places: the second product's factors have 1 and 2, so its product has 3 -- exactly the number of places the point moved, which is why the ratio is 1000.

Another way: Divide the two products directly: 900 divided by 0.900 is 1000. Slower, and it hides the reason.

Standardsmin grade 5

5.NBT.A.2 Explain patterns in number of zeros and placement of decimal point — Reading each decimal shift as a division by a power of ten, and combining them.
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Confirming the two products directly.

💡Takeaway. Moving the point in each factor moves it in the answer too, and the moves add up -- so you can tell the ratio without multiplying.

Variant 3 easy

How many times as large as $ⓑ$ is $ⓐ$ ?

$ⓐ$ $48 \times 17$
$ⓑ$ $0.48 \times 1.7$

Your answer

Show solution

1 · Understandwhat's really being asked

Two products use the same digits. The first multiplies 48 by 17; the second multiplies 0.48 by 1.7, the same digits with the decimal point moved. We must say how many times bigger the first is.

Givens

The first: 48 times 17.
The second: 0.48 times 1.7.
Both use the same digits; only the decimal points differ.

Unknowns

How many times as large the first product is as the second.

Constraints

Neither product needs to be worked out to compare them.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem

Compare the two multiplications factor by factor. Each factor is smaller by a power of ten, and the effects combine, so the answer follows from the shifts alone.

3 · Execute3 carry out the plan

1Compare each factor

#5 Look for a Pattern 5.NBT.A.2

0.48 is 48 with the point moved 2 places, so it is 100 times smaller. Likewise 1.7 is 10 times smaller than 17.

0.48 = 48 \div 100,\quad 1.7 = 17 \div 10

Each factor shrank by its own power of ten.

2See how the product changes

#9 Solve an Easier Related Problem 5.NBT.A.2

Shrinking one factor 100 times and the other 10 times shrinks the product by both together.

100 \times 10 = 1000

The decimal point in the answer moves 3 places, so the first product is 1000 times the second.

3Check by direct calculation

#5 Look for a Pattern 5.NBT.B.7

Working both out confirms it: the digits are identical and only the point has moved 3 places.

48 \times 17 = 816,\quad 0.48 \times 1.7 = 0.816

816 is 1000 times 0.816.

Answer: 1000 times

4 · Reviewdoes it hold up?

Count the decimal places: the second product's factors have 2 and 1, so its product has 3 -- exactly the number of places the point moved, which is why the ratio is 1000.

Another way: Divide the two products directly: 816 divided by 0.816 is 1000. Slower, and it hides the reason.

Standardsmin grade 5

5.NBT.A.2 Explain patterns in number of zeros and placement of decimal point — Reading each decimal shift as a division by a power of ten, and combining them.
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Confirming the two products directly.

💡Takeaway. Moving the point in each factor moves it in the answer too, and the moves add up -- so you can tell the ratio without multiplying.

Variant 4 easy

How many times as large as $ⓑ$ is $ⓐ$ ?

$ⓐ$ $28 \times 55$
$ⓑ$ $0.028 \times 0.55$

Your answer

Show solution

1 · Understandwhat's really being asked

Two products use the same digits. The first multiplies 28 by 55; the second multiplies 0.028 by 0.55, the same digits with the decimal point moved. We must say how many times bigger the first is.

Givens

The first: 28 times 55.
The second: 0.028 times 0.55.
Both use the same digits; only the decimal points differ.

Unknowns

How many times as large the first product is as the second.

Constraints

Neither product needs to be worked out to compare them.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem

Compare the two multiplications factor by factor. Each factor is smaller by a power of ten, and the effects combine, so the answer follows from the shifts alone.

3 · Execute3 carry out the plan

1Compare each factor

#5 Look for a Pattern 5.NBT.A.2

0.028 is 28 with the point moved 3 places, so it is 1000 times smaller. Likewise 0.55 is 100 times smaller than 55.

0.028 = 28 \div 1000,\quad 0.55 = 55 \div 100

Each factor shrank by its own power of ten.

2See how the product changes

#9 Solve an Easier Related Problem 5.NBT.A.2

Shrinking one factor 1000 times and the other 100 times shrinks the product by both together.

1000 \times 100 = 100000

The decimal point in the answer moves 5 places, so the first product is 100000 times the second.

3Check by direct calculation

#5 Look for a Pattern 5.NBT.B.7

Working both out confirms it: the digits are identical and only the point has moved 5 places.

28 \times 55 = 1540,\quad 0.028 \times 0.55 = 0.01540

1540 is 100000 times 0.01540.

Answer: 100000 times

4 · Reviewdoes it hold up?

Count the decimal places: the second product's factors have 3 and 2, so its product has 5 -- exactly the number of places the point moved, which is why the ratio is 100000.

Another way: Divide the two products directly: 1540 divided by 0.01540 is 100000. Slower, and it hides the reason.

Standardsmin grade 5

5.NBT.A.2 Explain patterns in number of zeros and placement of decimal point — Reading each decimal shift as a division by a power of ten, and combining them.
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Confirming the two products directly.

💡Takeaway. Moving the point in each factor moves it in the answer too, and the moves add up -- so you can tell the ratio without multiplying.

Variant 5 medium

How many times as large as $ⓑ$ is $ⓐ$ ?

$ⓐ$ $52 \times 63$
$ⓑ$ $5.2 \times 6.3$

Your answer

Show solution

1 · Understandwhat's really being asked

Two products use the same digits. The first multiplies 52 by 63; the second multiplies 5.2 by 6.3, the same digits with the decimal point moved. We must say how many times bigger the first is.

Givens

The first: 52 times 63.
The second: 5.2 times 6.3.
Both use the same digits; only the decimal points differ.

Unknowns

How many times as large the first product is as the second.

Constraints

Neither product needs to be worked out to compare them.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem

Compare the two multiplications factor by factor. Each factor is smaller by a power of ten, and the effects combine, so the answer follows from the shifts alone.

3 · Execute3 carry out the plan

1Compare each factor

#5 Look for a Pattern 5.NBT.A.2

5.2 is 52 with the point moved 1 place, so it is 10 times smaller. Likewise 6.3 is 10 times smaller than 63.

5.2 = 52 \div 10,\quad 6.3 = 63 \div 10

Each factor shrank by its own power of ten.

2See how the product changes

#9 Solve an Easier Related Problem 5.NBT.A.2

Shrinking one factor 10 times and the other 10 times shrinks the product by both together.

10 \times 10 = 100

The decimal point in the answer moves 2 places, so the first product is 100 times the second.

3Check by direct calculation

#5 Look for a Pattern 5.NBT.B.7

Working both out confirms it: the digits are identical and only the point has moved 2 places.

52 \times 63 = 3276,\quad 5.2 \times 6.3 = 32.76

3276 is 100 times 32.76.

Answer: 100 times

4 · Reviewdoes it hold up?

Count the decimal places: the second product's factors have 1 and 1, so its product has 2 -- exactly the number of places the point moved, which is why the ratio is 100.

Another way: Divide the two products directly: 3276 divided by 32.76 is 100. Slower, and it hides the reason.

Standardsmin grade 5

5.NBT.A.2 Explain patterns in number of zeros and placement of decimal point — Reading each decimal shift as a division by a power of ten, and combining them.
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Confirming the two products directly.

💡Takeaway. Moving the point in each factor moves it in the answer too, and the moves add up -- so you can tell the ratio without multiplying.

Variant 6 medium

How many times as large as $ⓑ$ is $ⓐ$ ?

$ⓐ$ $67 \times 43$
$ⓑ$ $0.67 \times 0.43$

Your answer

Show solution

1 · Understandwhat's really being asked

Two products use the same digits. The first multiplies 67 by 43; the second multiplies 0.67 by 0.43, the same digits with the decimal point moved. We must say how many times bigger the first is.

Givens

The first: 67 times 43.
The second: 0.67 times 0.43.
Both use the same digits; only the decimal points differ.

Unknowns

How many times as large the first product is as the second.

Constraints

Neither product needs to be worked out to compare them.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem

Compare the two multiplications factor by factor. Each factor is smaller by a power of ten, and the effects combine, so the answer follows from the shifts alone.

3 · Execute3 carry out the plan

1Compare each factor

#5 Look for a Pattern 5.NBT.A.2

0.67 is 67 with the point moved 2 places, so it is 100 times smaller. Likewise 0.43 is 100 times smaller than 43.

0.67 = 67 \div 100,\quad 0.43 = 43 \div 100

Each factor shrank by its own power of ten.

2See how the product changes

#9 Solve an Easier Related Problem 5.NBT.A.2

Shrinking one factor 100 times and the other 100 times shrinks the product by both together.

100 \times 100 = 10000

The decimal point in the answer moves 4 places, so the first product is 10000 times the second.

3Check by direct calculation

#5 Look for a Pattern 5.NBT.B.7

Working both out confirms it: the digits are identical and only the point has moved 4 places.

67 \times 43 = 2881,\quad 0.67 \times 0.43 = 0.2881

2881 is 10000 times 0.2881.

Answer: 10000 times

4 · Reviewdoes it hold up?

Count the decimal places: the second product's factors have 2 and 2, so its product has 4 -- exactly the number of places the point moved, which is why the ratio is 10000.

Another way: Divide the two products directly: 2881 divided by 0.2881 is 10000. Slower, and it hides the reason.

Standardsmin grade 5

5.NBT.A.2 Explain patterns in number of zeros and placement of decimal point — Reading each decimal shift as a division by a power of ten, and combining them.
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Confirming the two products directly.

💡Takeaway. Moving the point in each factor moves it in the answer too, and the moves add up -- so you can tell the ratio without multiplying.

Variant 7 medium

How many times as large as $ⓑ$ is $ⓐ$ ?

$ⓐ$ $75 \times 32$
$ⓑ$ $0.75 \times 0.032$

Your answer

Show solution

1 · Understandwhat's really being asked

Two products use the same digits. The first multiplies 75 by 32; the second multiplies 0.75 by 0.032, the same digits with the decimal point moved. We must say how many times bigger the first is.

Givens

The first: 75 times 32.
The second: 0.75 times 0.032.
Both use the same digits; only the decimal points differ.

Unknowns

How many times as large the first product is as the second.

Constraints

Neither product needs to be worked out to compare them.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem

Compare the two multiplications factor by factor. Each factor is smaller by a power of ten, and the effects combine, so the answer follows from the shifts alone.

3 · Execute3 carry out the plan

1Compare each factor

#5 Look for a Pattern 5.NBT.A.2

0.75 is 75 with the point moved 2 places, so it is 100 times smaller. Likewise 0.032 is 1000 times smaller than 32.

0.75 = 75 \div 100,\quad 0.032 = 32 \div 1000

Each factor shrank by its own power of ten.

2See how the product changes

#9 Solve an Easier Related Problem 5.NBT.A.2

Shrinking one factor 100 times and the other 1000 times shrinks the product by both together.

100 \times 1000 = 100000

The decimal point in the answer moves 5 places, so the first product is 100000 times the second.

3Check by direct calculation

#5 Look for a Pattern 5.NBT.B.7

Working both out confirms it: the digits are identical and only the point has moved 5 places.

75 \times 32 = 2400,\quad 0.75 \times 0.032 = 0.02400

2400 is 100000 times 0.02400.

Answer: 100000 times

4 · Reviewdoes it hold up?

Count the decimal places: the second product's factors have 2 and 3, so its product has 5 -- exactly the number of places the point moved, which is why the ratio is 100000.

Another way: Divide the two products directly: 2400 divided by 0.02400 is 100000. Slower, and it hides the reason.

Standardsmin grade 5

5.NBT.A.2 Explain patterns in number of zeros and placement of decimal point — Reading each decimal shift as a division by a power of ten, and combining them.
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Confirming the two products directly.

💡Takeaway. Moving the point in each factor moves it in the answer too, and the moves add up -- so you can tell the ratio without multiplying.

Variant 8 medium

How many times as large as $ⓑ$ is $ⓐ$ ?

$ⓐ$ $33 \times 78$
$ⓑ$ $0.033 \times 0.078$

Your answer

Show solution

1 · Understandwhat's really being asked

Two products use the same digits. The first multiplies 33 by 78; the second multiplies 0.033 by 0.078, the same digits with the decimal point moved. We must say how many times bigger the first is.

Givens

The first: 33 times 78.
The second: 0.033 times 0.078.
Both use the same digits; only the decimal points differ.

Unknowns

How many times as large the first product is as the second.

Constraints

Neither product needs to be worked out to compare them.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem

Compare the two multiplications factor by factor. Each factor is smaller by a power of ten, and the effects combine, so the answer follows from the shifts alone.

3 · Execute3 carry out the plan

1Compare each factor

#5 Look for a Pattern 5.NBT.A.2

0.033 is 33 with the point moved 3 places, so it is 1000 times smaller. Likewise 0.078 is 1000 times smaller than 78.

0.033 = 33 \div 1000,\quad 0.078 = 78 \div 1000

Each factor shrank by its own power of ten.

2See how the product changes

#9 Solve an Easier Related Problem 5.NBT.A.2

Shrinking one factor 1000 times and the other 1000 times shrinks the product by both together.

1000 \times 1000 = 1000000

The decimal point in the answer moves 6 places, so the first product is 1000000 times the second.

3Check by direct calculation

#5 Look for a Pattern 5.NBT.B.7

Working both out confirms it: the digits are identical and only the point has moved 6 places.

33 \times 78 = 2574,\quad 0.033 \times 0.078 = 0.002574

2574 is 1000000 times 0.002574.

Answer: 1000000 times

4 · Reviewdoes it hold up?

Count the decimal places: the second product's factors have 3 and 3, so its product has 6 -- exactly the number of places the point moved, which is why the ratio is 1000000.

Another way: Divide the two products directly: 2574 divided by 0.002574 is 1000000. Slower, and it hides the reason.

Standardsmin grade 5

5.NBT.A.2 Explain patterns in number of zeros and placement of decimal point — Reading each decimal shift as a division by a power of ten, and combining them.
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Confirming the two products directly.

💡Takeaway. Moving the point in each factor moves it in the answer too, and the moves add up -- so you can tell the ratio without multiplying.

Variant 9 hard

How many times as large as $ⓑ$ is $ⓐ$ ?

$ⓐ$ $81 \times 59$
$ⓑ$ $0.81 \times 5.9$

Your answer

Show solution

1 · Understandwhat's really being asked

Two products use the same digits. The first multiplies 81 by 59; the second multiplies 0.81 by 5.9, the same digits with the decimal point moved. We must say how many times bigger the first is.

Givens

The first: 81 times 59.
The second: 0.81 times 5.9.
Both use the same digits; only the decimal points differ.

Unknowns

How many times as large the first product is as the second.

Constraints

Neither product needs to be worked out to compare them.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem

Compare the two multiplications factor by factor. Each factor is smaller by a power of ten, and the effects combine, so the answer follows from the shifts alone.

3 · Execute3 carry out the plan

1Compare each factor

#5 Look for a Pattern 5.NBT.A.2

0.81 is 81 with the point moved 2 places, so it is 100 times smaller. Likewise 5.9 is 10 times smaller than 59.

0.81 = 81 \div 100,\quad 5.9 = 59 \div 10

Each factor shrank by its own power of ten.

2See how the product changes

#9 Solve an Easier Related Problem 5.NBT.A.2

Shrinking one factor 100 times and the other 10 times shrinks the product by both together.

100 \times 10 = 1000

The decimal point in the answer moves 3 places, so the first product is 1000 times the second.

3Check by direct calculation

#5 Look for a Pattern 5.NBT.B.7

Working both out confirms it: the digits are identical and only the point has moved 3 places.

81 \times 59 = 4779,\quad 0.81 \times 5.9 = 4.779

4779 is 1000 times 4.779.

Answer: 1000 times

4 · Reviewdoes it hold up?

Count the decimal places: the second product's factors have 2 and 1, so its product has 3 -- exactly the number of places the point moved, which is why the ratio is 1000.

Another way: Divide the two products directly: 4779 divided by 4.779 is 1000. Slower, and it hides the reason.

Standardsmin grade 5

5.NBT.A.2 Explain patterns in number of zeros and placement of decimal point — Reading each decimal shift as a division by a power of ten, and combining them.
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Confirming the two products directly.

💡Takeaway. Moving the point in each factor moves it in the answer too, and the moves add up -- so you can tell the ratio without multiplying.

Variant 10 hard

How many times as large as $ⓑ$ is $ⓐ$ ?

$ⓐ$ $19 \times 84$
$ⓑ$ $0.019 \times 8.4$

Your answer

Show solution

1 · Understandwhat's really being asked

Two products use the same digits. The first multiplies 19 by 84; the second multiplies 0.019 by 8.4, the same digits with the decimal point moved. We must say how many times bigger the first is.

Givens

The first: 19 times 84.
The second: 0.019 times 8.4.
Both use the same digits; only the decimal points differ.

Unknowns

How many times as large the first product is as the second.

Constraints

Neither product needs to be worked out to compare them.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem

Compare the two multiplications factor by factor. Each factor is smaller by a power of ten, and the effects combine, so the answer follows from the shifts alone.

3 · Execute3 carry out the plan

1Compare each factor

#5 Look for a Pattern 5.NBT.A.2

0.019 is 19 with the point moved 3 places, so it is 1000 times smaller. Likewise 8.4 is 10 times smaller than 84.

0.019 = 19 \div 1000,\quad 8.4 = 84 \div 10

Each factor shrank by its own power of ten.

2See how the product changes

#9 Solve an Easier Related Problem 5.NBT.A.2

Shrinking one factor 1000 times and the other 10 times shrinks the product by both together.

1000 \times 10 = 10000

The decimal point in the answer moves 4 places, so the first product is 10000 times the second.

3Check by direct calculation

#5 Look for a Pattern 5.NBT.B.7

Working both out confirms it: the digits are identical and only the point has moved 4 places.

19 \times 84 = 1596,\quad 0.019 \times 8.4 = 0.1596

1596 is 10000 times 0.1596.

Answer: 10000 times

4 · Reviewdoes it hold up?

Count the decimal places: the second product's factors have 3 and 1, so its product has 4 -- exactly the number of places the point moved, which is why the ratio is 10000.

Another way: Divide the two products directly: 1596 divided by 0.1596 is 10000. Slower, and it hides the reason.

Standardsmin grade 5

5.NBT.A.2 Explain patterns in number of zeros and placement of decimal point — Reading each decimal shift as a division by a power of ten, and combining them.
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Confirming the two products directly.

💡Takeaway. Moving the point in each factor moves it in the answer too, and the moves add up -- so you can tell the ratio without multiplying.

Variant 11 hard

How many times as large as $ⓑ$ is $ⓐ$ ?

$ⓐ$ $46 \times 91$
$ⓑ$ $4.6 \times 0.091$

Your answer

Show solution

1 · Understandwhat's really being asked

Two products use the same digits. The first multiplies 46 by 91; the second multiplies 4.6 by 0.091, the same digits with the decimal point moved. We must say how many times bigger the first is.

Givens

The first: 46 times 91.
The second: 4.6 times 0.091.
Both use the same digits; only the decimal points differ.

Unknowns

How many times as large the first product is as the second.

Constraints

Neither product needs to be worked out to compare them.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem

Compare the two multiplications factor by factor. Each factor is smaller by a power of ten, and the effects combine, so the answer follows from the shifts alone.

3 · Execute3 carry out the plan

1Compare each factor

#5 Look for a Pattern 5.NBT.A.2

4.6 is 46 with the point moved 1 place, so it is 10 times smaller. Likewise 0.091 is 1000 times smaller than 91.

4.6 = 46 \div 10,\quad 0.091 = 91 \div 1000

Each factor shrank by its own power of ten.

2See how the product changes

#9 Solve an Easier Related Problem 5.NBT.A.2

Shrinking one factor 10 times and the other 1000 times shrinks the product by both together.

10 \times 1000 = 10000

The decimal point in the answer moves 4 places, so the first product is 10000 times the second.

3Check by direct calculation

#5 Look for a Pattern 5.NBT.B.7

Working both out confirms it: the digits are identical and only the point has moved 4 places.

46 \times 91 = 4186,\quad 4.6 \times 0.091 = 0.4186

4186 is 10000 times 0.4186.

Answer: 10000 times

4 · Reviewdoes it hold up?

Count the decimal places: the second product's factors have 1 and 3, so its product has 4 -- exactly the number of places the point moved, which is why the ratio is 10000.

Another way: Divide the two products directly: 4186 divided by 0.4186 is 10000. Slower, and it hides the reason.

Standardsmin grade 5

5.NBT.A.2 Explain patterns in number of zeros and placement of decimal point — Reading each decimal shift as a division by a power of ten, and combining them.
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Confirming the two products directly.

💡Takeaway. Moving the point in each factor moves it in the answer too, and the moves add up -- so you can tell the ratio without multiplying.

Variant 12 hard

How many times as large as $ⓑ$ is $ⓐ$ ?

$ⓐ$ $94 \times 26$
$ⓑ$ $9.4 \times 0.26$

Your answer

Show solution

1 · Understandwhat's really being asked

Two products use the same digits. The first multiplies 94 by 26; the second multiplies 9.4 by 0.26, the same digits with the decimal point moved. We must say how many times bigger the first is.

Givens

The first: 94 times 26.
The second: 9.4 times 0.26.
Both use the same digits; only the decimal points differ.

Unknowns

How many times as large the first product is as the second.

Constraints

Neither product needs to be worked out to compare them.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem

Compare the two multiplications factor by factor. Each factor is smaller by a power of ten, and the effects combine, so the answer follows from the shifts alone.

3 · Execute3 carry out the plan

1Compare each factor

#5 Look for a Pattern 5.NBT.A.2

9.4 is 94 with the point moved 1 place, so it is 10 times smaller. Likewise 0.26 is 100 times smaller than 26.

9.4 = 94 \div 10,\quad 0.26 = 26 \div 100

Each factor shrank by its own power of ten.

2See how the product changes

#9 Solve an Easier Related Problem 5.NBT.A.2

Shrinking one factor 10 times and the other 100 times shrinks the product by both together.

10 \times 100 = 1000

The decimal point in the answer moves 3 places, so the first product is 1000 times the second.

3Check by direct calculation

#5 Look for a Pattern 5.NBT.B.7

Working both out confirms it: the digits are identical and only the point has moved 3 places.

94 \times 26 = 2444,\quad 9.4 \times 0.26 = 2.444

2444 is 1000 times 2.444.

Answer: 1000 times

4 · Reviewdoes it hold up?

Count the decimal places: the second product's factors have 1 and 2, so its product has 3 -- exactly the number of places the point moved, which is why the ratio is 1000.

Another way: Divide the two products directly: 2444 divided by 2.444 is 1000. Slower, and it hides the reason.

Standardsmin grade 5

5.NBT.A.2 Explain patterns in number of zeros and placement of decimal point — Reading each decimal shift as a division by a power of ten, and combining them.
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Confirming the two products directly.

💡Takeaway. Moving the point in each factor moves it in the answer too, and the moves add up -- so you can tell the ratio without multiplying.

Decimal point position predicts the product · 12 practice problems

Generated variants — 12

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Where next?