← Decimal point position predicts the product · Place-Value Regrouping

Decimal point position predicts the product · 12 practice problems

5.NBT.A.25.NBT.B.7

Generated variants — 12

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 easy answer: 10000 times

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

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

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 answer: 1000 times

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

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

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 answer: 1000 times

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

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

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 answer: 100000 times

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

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

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 answer: 100 times

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

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

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 answer: 10000 times

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

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

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 answer: 100000 times

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

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

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 answer: 1000000 times

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

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

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 answer: 1000 times

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

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

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 answer: 10000 times

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

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

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 answer: 10000 times

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

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

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 answer: 1000 times

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

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

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.