Practice · Factor out a common factor with parentheses · Sensim Math

Variant 1 easy

Rewrite the following as a simpler expression, then evaluate it.

$125 \times 7 + 12.5 \times 6 + 1.25 \times 30$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Three products are added. Their first factors -- 125, 12.5 and 1.25 -- are the same digits with the decimal point in different places. We must simplify before evaluating.

Givens

The expression is 125 x 7 + 12.5 x 6 + 1.25 x 30.
The three first factors share their digits.

Unknowns

A simpler equivalent expression, and its value.

Constraints

The rewrite must keep the same value, not just a similar shape.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #5 Look for a Pattern#7 Identify Subproblems

Nothing here is a common factor as written. But every first factor is 1.25 times a power of ten, so moving that power onto the other side of each product makes 1.25 common to all three.

3 · Execute4 carry out the plan

1Rewrite each decimal as a multiple of 1.25

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

125 is 1.25 times 100, 12.5 is 1.25 times 10, and 1.25 is 1.25 itself.

125 = 1.25 \times 100,\quad 12.5 = 1.25 \times 10

The same digits, three places apart.

2Turn each product into 1.25 times a number

#7 Identify Subproblems 3.OA.B.5

Multiplication can be regrouped, so the power of ten joins the other factor.

1.25 \times 700 + 1.25 \times 60 + 1.25 \times 30

Now 1.25 appears in every term.

3Factor out 1.25 with parentheses

#15 Organize Information in More Ways 5.OA.A.1

The distributive law runs backwards here: a common factor comes out and the rest goes inside brackets.

1.25 \times (700 + 60 + 30)

Three multiplications have become one.

4Add inside the parentheses, then multiply

#15 Organize Information in More Ways 5.NBT.B.7

The bracket is plain whole-number addition, and the final multiplication is by a round number.

1.25 \times 790 = 987.5

The value is 987.5.

Answer:

1.25 \times (700 + 60 + 30) = 987.5

4 · Reviewdoes it hold up?

Evaluate the original directly as a check: 875 + 75 + 37.5 = 987.5. The rewrite kept the value.

Another way: Factoring out 125 instead works too, but leaves fractions of a hundredth inside the brackets; taking the smallest of the three keeps every number whole.

Standardsmin grade 5

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate — Writing the factored form with parentheses and evaluating it.
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Relating the three decimals by powers of ten and finishing the multiplication.
3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Regrouping each product so the shared factor can come out.

💡Takeaway. Numbers that look different can share a factor once you move the decimal point's worth onto the other side.

Variant 2 easy

Rewrite the following as a simpler expression, then evaluate it.

$168 \times 6 + 16.8 \times 4 + 1.68 \times 25$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Three products are added. Their first factors -- 168, 16.8 and 1.68 -- are the same digits with the decimal point in different places. We must simplify before evaluating.

Givens

The expression is 168 x 6 + 16.8 x 4 + 1.68 x 25.
The three first factors share their digits.

Unknowns

A simpler equivalent expression, and its value.

Constraints

The rewrite must keep the same value, not just a similar shape.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #5 Look for a Pattern#7 Identify Subproblems

Nothing here is a common factor as written. But every first factor is 1.68 times a power of ten, so moving that power onto the other side of each product makes 1.68 common to all three.

3 · Execute4 carry out the plan

1Rewrite each decimal as a multiple of 1.68

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

168 is 1.68 times 100, 16.8 is 1.68 times 10, and 1.68 is 1.68 itself.

168 = 1.68 \times 100,\quad 16.8 = 1.68 \times 10

The same digits, three places apart.

2Turn each product into 1.68 times a number

#7 Identify Subproblems 3.OA.B.5

Multiplication can be regrouped, so the power of ten joins the other factor.

1.68 \times 600 + 1.68 \times 40 + 1.68 \times 25

Now 1.68 appears in every term.

3Factor out 1.68 with parentheses

#15 Organize Information in More Ways 5.OA.A.1

The distributive law runs backwards here: a common factor comes out and the rest goes inside brackets.

1.68 \times (600 + 40 + 25)

Three multiplications have become one.

4Add inside the parentheses, then multiply

#15 Organize Information in More Ways 5.NBT.B.7

The bracket is plain whole-number addition, and the final multiplication is by a round number.

1.68 \times 665 = 1117.2

The value is 1117.2.

Answer:

1.68 \times (600 + 40 + 25) = 1117.2

4 · Reviewdoes it hold up?

Evaluate the original directly as a check: 1008 + 67.2 + 42 = 1117.2. The rewrite kept the value.

Another way: Factoring out 168 instead works too, but leaves fractions of a hundredth inside the brackets; taking the smallest of the three keeps every number whole.

Standardsmin grade 5

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate — Writing the factored form with parentheses and evaluating it.
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Relating the three decimals by powers of ten and finishing the multiplication.
3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Regrouping each product so the shared factor can come out.

💡Takeaway. Numbers that look different can share a factor once you move the decimal point's worth onto the other side.

Variant 3 easy

Rewrite the following as a simpler expression, then evaluate it.

$187.5 \times 9 + 18.75 \times 6 + 1.875 \times 15$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Three products are added. Their first factors -- 187.5, 18.75 and 1.875 -- are the same digits with the decimal point in different places. We must simplify before evaluating.

Givens

The expression is 187.5 x 9 + 18.75 x 6 + 1.875 x 15.
The three first factors share their digits.

Unknowns

A simpler equivalent expression, and its value.

Constraints

The rewrite must keep the same value, not just a similar shape.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #5 Look for a Pattern#7 Identify Subproblems

Nothing here is a common factor as written. But every first factor is 1.875 times a power of ten, so moving that power onto the other side of each product makes 1.875 common to all three.

3 · Execute4 carry out the plan

1Rewrite each decimal as a multiple of 1.875

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

187.5 is 1.875 times 100, 18.75 is 1.875 times 10, and 1.875 is 1.875 itself.

187.5 = 1.875 \times 100,\quad 18.75 = 1.875 \times 10

The same digits, three places apart.

2Turn each product into 1.875 times a number

#7 Identify Subproblems 3.OA.B.5

Multiplication can be regrouped, so the power of ten joins the other factor.

1.875 \times 900 + 1.875 \times 60 + 1.875 \times 15

Now 1.875 appears in every term.

3Factor out 1.875 with parentheses

#15 Organize Information in More Ways 5.OA.A.1

The distributive law runs backwards here: a common factor comes out and the rest goes inside brackets.

1.875 \times (900 + 60 + 15)

Three multiplications have become one.

4Add inside the parentheses, then multiply

#15 Organize Information in More Ways 5.NBT.B.7

The bracket is plain whole-number addition, and the final multiplication is by a round number.

1.875 \times 975 = 1828.125

The value is 1828.125.

Answer:

1.875 \times (900 + 60 + 15) = 1828.125

4 · Reviewdoes it hold up?

Evaluate the original directly as a check: 1687.5 + 112.5 + 28.125 = 1828.125. The rewrite kept the value.

Another way: Factoring out 187.5 instead works too, but leaves fractions of a hundredth inside the brackets; taking the smallest of the three keeps every number whole.

Standardsmin grade 5

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate — Writing the factored form with parentheses and evaluating it.
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Relating the three decimals by powers of ten and finishing the multiplication.
3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Regrouping each product so the shared factor can come out.

💡Takeaway. Numbers that look different can share a factor once you move the decimal point's worth onto the other side.

Variant 4 easy

Rewrite the following as a simpler expression, then evaluate it.

$217.5 \times 8 + 21.75 \times 5 + 2.175 \times 40$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Three products are added. Their first factors -- 217.5, 21.75 and 2.175 -- are the same digits with the decimal point in different places. We must simplify before evaluating.

Givens

The expression is 217.5 x 8 + 21.75 x 5 + 2.175 x 40.
The three first factors share their digits.

Unknowns

A simpler equivalent expression, and its value.

Constraints

The rewrite must keep the same value, not just a similar shape.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #5 Look for a Pattern#7 Identify Subproblems

Nothing here is a common factor as written. But every first factor is 2.175 times a power of ten, so moving that power onto the other side of each product makes 2.175 common to all three.

3 · Execute4 carry out the plan

1Rewrite each decimal as a multiple of 2.175

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

217.5 is 2.175 times 100, 21.75 is 2.175 times 10, and 2.175 is 2.175 itself.

217.5 = 2.175 \times 100,\quad 21.75 = 2.175 \times 10

The same digits, three places apart.

2Turn each product into 2.175 times a number

#7 Identify Subproblems 3.OA.B.5

Multiplication can be regrouped, so the power of ten joins the other factor.

2.175 \times 800 + 2.175 \times 50 + 2.175 \times 40

Now 2.175 appears in every term.

3Factor out 2.175 with parentheses

#15 Organize Information in More Ways 5.OA.A.1

The distributive law runs backwards here: a common factor comes out and the rest goes inside brackets.

2.175 \times (800 + 50 + 40)

Three multiplications have become one.

4Add inside the parentheses, then multiply

#15 Organize Information in More Ways 5.NBT.B.7

The bracket is plain whole-number addition, and the final multiplication is by a round number.

2.175 \times 890 = 1935.75

The value is 1935.75.

Answer:

2.175 \times (800 + 50 + 40) = 1935.75

4 · Reviewdoes it hold up?

Evaluate the original directly as a check: 1740 + 108.75 + 87 = 1935.75. The rewrite kept the value.

Another way: Factoring out 217.5 instead works too, but leaves fractions of a hundredth inside the brackets; taking the smallest of the three keeps every number whole.

Standardsmin grade 5

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate — Writing the factored form with parentheses and evaluating it.
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Relating the three decimals by powers of ten and finishing the multiplication.
3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Regrouping each product so the shared factor can come out.

💡Takeaway. Numbers that look different can share a factor once you move the decimal point's worth onto the other side.

Variant 5 medium

Rewrite the following as a simpler expression, then evaluate it.

$264 \times 8 + 26.4 \times 7 + 2.64 \times 20$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Three products are added. Their first factors -- 264, 26.4 and 2.64 -- are the same digits with the decimal point in different places. We must simplify before evaluating.

Givens

The expression is 264 x 8 + 26.4 x 7 + 2.64 x 20.
The three first factors share their digits.

Unknowns

A simpler equivalent expression, and its value.

Constraints

The rewrite must keep the same value, not just a similar shape.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #5 Look for a Pattern#7 Identify Subproblems

Nothing here is a common factor as written. But every first factor is 2.64 times a power of ten, so moving that power onto the other side of each product makes 2.64 common to all three.

3 · Execute4 carry out the plan

1Rewrite each decimal as a multiple of 2.64

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

264 is 2.64 times 100, 26.4 is 2.64 times 10, and 2.64 is 2.64 itself.

264 = 2.64 \times 100,\quad 26.4 = 2.64 \times 10

The same digits, three places apart.

2Turn each product into 2.64 times a number

#7 Identify Subproblems 3.OA.B.5

Multiplication can be regrouped, so the power of ten joins the other factor.

2.64 \times 800 + 2.64 \times 70 + 2.64 \times 20

Now 2.64 appears in every term.

3Factor out 2.64 with parentheses

#15 Organize Information in More Ways 5.OA.A.1

The distributive law runs backwards here: a common factor comes out and the rest goes inside brackets.

2.64 \times (800 + 70 + 20)

Three multiplications have become one.

4Add inside the parentheses, then multiply

#15 Organize Information in More Ways 5.NBT.B.7

The bracket is plain whole-number addition, and the final multiplication is by a round number.

2.64 \times 890 = 2349.6

The value is 2349.6.

Answer:

2.64 \times (800 + 70 + 20) = 2349.6

4 · Reviewdoes it hold up?

Evaluate the original directly as a check: 2112 + 184.8 + 52.8 = 2349.6. The rewrite kept the value.

Another way: Factoring out 264 instead works too, but leaves fractions of a hundredth inside the brackets; taking the smallest of the three keeps every number whole.

Standardsmin grade 5

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate — Writing the factored form with parentheses and evaluating it.
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Relating the three decimals by powers of ten and finishing the multiplication.
3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Regrouping each product so the shared factor can come out.

💡Takeaway. Numbers that look different can share a factor once you move the decimal point's worth onto the other side.

Variant 6 medium

Rewrite the following as a simpler expression, then evaluate it.

$292.5 \times 5 + 29.25 \times 8 + 2.925 \times 35$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Three products are added. Their first factors -- 292.5, 29.25 and 2.925 -- are the same digits with the decimal point in different places. We must simplify before evaluating.

Givens

The expression is 292.5 x 5 + 29.25 x 8 + 2.925 x 35.
The three first factors share their digits.

Unknowns

A simpler equivalent expression, and its value.

Constraints

The rewrite must keep the same value, not just a similar shape.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #5 Look for a Pattern#7 Identify Subproblems

Nothing here is a common factor as written. But every first factor is 2.925 times a power of ten, so moving that power onto the other side of each product makes 2.925 common to all three.

3 · Execute4 carry out the plan

1Rewrite each decimal as a multiple of 2.925

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

292.5 is 2.925 times 100, 29.25 is 2.925 times 10, and 2.925 is 2.925 itself.

292.5 = 2.925 \times 100,\quad 29.25 = 2.925 \times 10

The same digits, three places apart.

2Turn each product into 2.925 times a number

#7 Identify Subproblems 3.OA.B.5

Multiplication can be regrouped, so the power of ten joins the other factor.

2.925 \times 500 + 2.925 \times 80 + 2.925 \times 35

Now 2.925 appears in every term.

3Factor out 2.925 with parentheses

#15 Organize Information in More Ways 5.OA.A.1

The distributive law runs backwards here: a common factor comes out and the rest goes inside brackets.

2.925 \times (500 + 80 + 35)

Three multiplications have become one.

4Add inside the parentheses, then multiply

#15 Organize Information in More Ways 5.NBT.B.7

The bracket is plain whole-number addition, and the final multiplication is by a round number.

2.925 \times 615 = 1798.875

The value is 1798.875.

Answer:

2.925 \times (500 + 80 + 35) = 1798.875

4 · Reviewdoes it hold up?

Evaluate the original directly as a check: 1462.5 + 234 + 102.375 = 1798.875. The rewrite kept the value.

Another way: Factoring out 292.5 instead works too, but leaves fractions of a hundredth inside the brackets; taking the smallest of the three keeps every number whole.

Standardsmin grade 5

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate — Writing the factored form with parentheses and evaluating it.
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Relating the three decimals by powers of ten and finishing the multiplication.
3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Regrouping each product so the shared factor can come out.

💡Takeaway. Numbers that look different can share a factor once you move the decimal point's worth onto the other side.

Variant 7 medium

Rewrite the following as a simpler expression, then evaluate it.

$312.5 \times 7 + 31.25 \times 3 + 3.125 \times 45$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Three products are added. Their first factors -- 312.5, 31.25 and 3.125 -- are the same digits with the decimal point in different places. We must simplify before evaluating.

Givens

The expression is 312.5 x 7 + 31.25 x 3 + 3.125 x 45.
The three first factors share their digits.

Unknowns

A simpler equivalent expression, and its value.

Constraints

The rewrite must keep the same value, not just a similar shape.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #5 Look for a Pattern#7 Identify Subproblems

Nothing here is a common factor as written. But every first factor is 3.125 times a power of ten, so moving that power onto the other side of each product makes 3.125 common to all three.

3 · Execute4 carry out the plan

1Rewrite each decimal as a multiple of 3.125

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

312.5 is 3.125 times 100, 31.25 is 3.125 times 10, and 3.125 is 3.125 itself.

312.5 = 3.125 \times 100,\quad 31.25 = 3.125 \times 10

The same digits, three places apart.

2Turn each product into 3.125 times a number

#7 Identify Subproblems 3.OA.B.5

Multiplication can be regrouped, so the power of ten joins the other factor.

3.125 \times 700 + 3.125 \times 30 + 3.125 \times 45

Now 3.125 appears in every term.

3Factor out 3.125 with parentheses

#15 Organize Information in More Ways 5.OA.A.1

The distributive law runs backwards here: a common factor comes out and the rest goes inside brackets.

3.125 \times (700 + 30 + 45)

Three multiplications have become one.

4Add inside the parentheses, then multiply

#15 Organize Information in More Ways 5.NBT.B.7

The bracket is plain whole-number addition, and the final multiplication is by a round number.

3.125 \times 775 = 2421.875

The value is 2421.875.

Answer:

3.125 \times (700 + 30 + 45) = 2421.875

4 · Reviewdoes it hold up?

Evaluate the original directly as a check: 2187.5 + 93.75 + 140.625 = 2421.875. The rewrite kept the value.

Another way: Factoring out 312.5 instead works too, but leaves fractions of a hundredth inside the brackets; taking the smallest of the three keeps every number whole.

Standardsmin grade 5

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate — Writing the factored form with parentheses and evaluating it.
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Relating the three decimals by powers of ten and finishing the multiplication.
3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Regrouping each product so the shared factor can come out.

💡Takeaway. Numbers that look different can share a factor once you move the decimal point's worth onto the other side.

Variant 8 medium

Rewrite the following as a simpler expression, then evaluate it.

$360 \times 4 + 36 \times 9 + 3.6 \times 50$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Three products are added. Their first factors -- 360, 36 and 3.6 -- are the same digits with the decimal point in different places. We must simplify before evaluating.

Givens

The expression is 360 x 4 + 36 x 9 + 3.6 x 50.
The three first factors share their digits.

Unknowns

A simpler equivalent expression, and its value.

Constraints

The rewrite must keep the same value, not just a similar shape.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #5 Look for a Pattern#7 Identify Subproblems

Nothing here is a common factor as written. But every first factor is 3.6 times a power of ten, so moving that power onto the other side of each product makes 3.6 common to all three.

3 · Execute4 carry out the plan

1Rewrite each decimal as a multiple of 3.6

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

360 is 3.6 times 100, 36 is 3.6 times 10, and 3.6 is 3.6 itself.

360 = 3.6 \times 100,\quad 36 = 3.6 \times 10

The same digits, three places apart.

2Turn each product into 3.6 times a number

#7 Identify Subproblems 3.OA.B.5

Multiplication can be regrouped, so the power of ten joins the other factor.

3.6 \times 400 + 3.6 \times 90 + 3.6 \times 50

Now 3.6 appears in every term.

3Factor out 3.6 with parentheses

#15 Organize Information in More Ways 5.OA.A.1

The distributive law runs backwards here: a common factor comes out and the rest goes inside brackets.

3.6 \times (400 + 90 + 50)

Three multiplications have become one.

4Add inside the parentheses, then multiply

#15 Organize Information in More Ways 5.NBT.B.7

The bracket is plain whole-number addition, and the final multiplication is by a round number.

3.6 \times 540 = 1944

The value is 1944.

Answer:

3.6 \times (400 + 90 + 50) = 1944

4 · Reviewdoes it hold up?

Evaluate the original directly as a check: 1440 + 324 + 180 = 1944. The rewrite kept the value.

Another way: Factoring out 360 instead works too, but leaves fractions of a hundredth inside the brackets; taking the smallest of the three keeps every number whole.

Standardsmin grade 5

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate — Writing the factored form with parentheses and evaluating it.
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Relating the three decimals by powers of ten and finishing the multiplication.
3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Regrouping each product so the shared factor can come out.

💡Takeaway. Numbers that look different can share a factor once you move the decimal point's worth onto the other side.

Variant 9 hard

Rewrite the following as a simpler expression, then evaluate it.

$423.4 \times 9 + 42.34 \times 8 + 4.234 \times 20$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Three products are added. Their first factors -- 423.4, 42.34 and 4.234 -- are the same digits with the decimal point in different places. We must simplify before evaluating.

Givens

The expression is 423.4 x 9 + 42.34 x 8 + 4.234 x 20.
The three first factors share their digits.

Unknowns

A simpler equivalent expression, and its value.

Constraints

The rewrite must keep the same value, not just a similar shape.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #5 Look for a Pattern#7 Identify Subproblems

Nothing here is a common factor as written. But every first factor is 4.234 times a power of ten, so moving that power onto the other side of each product makes 4.234 common to all three.

3 · Execute4 carry out the plan

1Rewrite each decimal as a multiple of 4.234

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

423.4 is 4.234 times 100, 42.34 is 4.234 times 10, and 4.234 is 4.234 itself.

423.4 = 4.234 \times 100,\quad 42.34 = 4.234 \times 10

The same digits, three places apart.

2Turn each product into 4.234 times a number

#7 Identify Subproblems 3.OA.B.5

Multiplication can be regrouped, so the power of ten joins the other factor.

4.234 \times 900 + 4.234 \times 80 + 4.234 \times 20

Now 4.234 appears in every term.

3Factor out 4.234 with parentheses

#15 Organize Information in More Ways 5.OA.A.1

The distributive law runs backwards here: a common factor comes out and the rest goes inside brackets.

4.234 \times (900 + 80 + 20)

Three multiplications have become one.

4Add inside the parentheses, then multiply

#15 Organize Information in More Ways 5.NBT.B.7

The bracket is plain whole-number addition, and the final multiplication is by a round number.

4.234 \times 1000 = 4234

The value is 4234.

Answer:

4.234 \times (900 + 80 + 20) = 4234

4 · Reviewdoes it hold up?

Evaluate the original directly as a check: 3810.6 + 338.72 + 84.68 = 4234. The rewrite kept the value.

Another way: Factoring out 423.4 instead works too, but leaves fractions of a hundredth inside the brackets; taking the smallest of the three keeps every number whole.

Standardsmin grade 5

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate — Writing the factored form with parentheses and evaluating it.
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Relating the three decimals by powers of ten and finishing the multiplication.
3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Regrouping each product so the shared factor can come out.

💡Takeaway. Numbers that look different can share a factor once you move the decimal point's worth onto the other side.

Variant 10 hard

Rewrite the following as a simpler expression, then evaluate it.

$440 \times 2 + 44 \times 9 + 4.4 \times 70$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Three products are added. Their first factors -- 440, 44 and 4.4 -- are the same digits with the decimal point in different places. We must simplify before evaluating.

Givens

The expression is 440 x 2 + 44 x 9 + 4.4 x 70.
The three first factors share their digits.

Unknowns

A simpler equivalent expression, and its value.

Constraints

The rewrite must keep the same value, not just a similar shape.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #5 Look for a Pattern#7 Identify Subproblems

Nothing here is a common factor as written. But every first factor is 4.4 times a power of ten, so moving that power onto the other side of each product makes 4.4 common to all three.

3 · Execute4 carry out the plan

1Rewrite each decimal as a multiple of 4.4

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

440 is 4.4 times 100, 44 is 4.4 times 10, and 4.4 is 4.4 itself.

440 = 4.4 \times 100,\quad 44 = 4.4 \times 10

The same digits, three places apart.

2Turn each product into 4.4 times a number

#7 Identify Subproblems 3.OA.B.5

Multiplication can be regrouped, so the power of ten joins the other factor.

4.4 \times 200 + 4.4 \times 90 + 4.4 \times 70

Now 4.4 appears in every term.

3Factor out 4.4 with parentheses

#15 Organize Information in More Ways 5.OA.A.1

The distributive law runs backwards here: a common factor comes out and the rest goes inside brackets.

4.4 \times (200 + 90 + 70)

Three multiplications have become one.

4Add inside the parentheses, then multiply

#15 Organize Information in More Ways 5.NBT.B.7

The bracket is plain whole-number addition, and the final multiplication is by a round number.

4.4 \times 360 = 1584

The value is 1584.

Answer:

4.4 \times (200 + 90 + 70) = 1584

4 · Reviewdoes it hold up?

Evaluate the original directly as a check: 880 + 396 + 308 = 1584. The rewrite kept the value.

Another way: Factoring out 440 instead works too, but leaves fractions of a hundredth inside the brackets; taking the smallest of the three keeps every number whole.

Standardsmin grade 5

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate — Writing the factored form with parentheses and evaluating it.
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Relating the three decimals by powers of ten and finishing the multiplication.
3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Regrouping each product so the shared factor can come out.

💡Takeaway. Numbers that look different can share a factor once you move the decimal point's worth onto the other side.

Variant 11 hard

Rewrite the following as a simpler expression, then evaluate it.

$504 \times 3 + 50.4 \times 7 + 5.04 \times 60$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Three products are added. Their first factors -- 504, 50.4 and 5.04 -- are the same digits with the decimal point in different places. We must simplify before evaluating.

Givens

The expression is 504 x 3 + 50.4 x 7 + 5.04 x 60.
The three first factors share their digits.

Unknowns

A simpler equivalent expression, and its value.

Constraints

The rewrite must keep the same value, not just a similar shape.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #5 Look for a Pattern#7 Identify Subproblems

Nothing here is a common factor as written. But every first factor is 5.04 times a power of ten, so moving that power onto the other side of each product makes 5.04 common to all three.

3 · Execute4 carry out the plan

1Rewrite each decimal as a multiple of 5.04

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

504 is 5.04 times 100, 50.4 is 5.04 times 10, and 5.04 is 5.04 itself.

504 = 5.04 \times 100,\quad 50.4 = 5.04 \times 10

The same digits, three places apart.

2Turn each product into 5.04 times a number

#7 Identify Subproblems 3.OA.B.5

Multiplication can be regrouped, so the power of ten joins the other factor.

5.04 \times 300 + 5.04 \times 70 + 5.04 \times 60

Now 5.04 appears in every term.

3Factor out 5.04 with parentheses

#15 Organize Information in More Ways 5.OA.A.1

The distributive law runs backwards here: a common factor comes out and the rest goes inside brackets.

5.04 \times (300 + 70 + 60)

Three multiplications have become one.

4Add inside the parentheses, then multiply

#15 Organize Information in More Ways 5.NBT.B.7

The bracket is plain whole-number addition, and the final multiplication is by a round number.

5.04 \times 430 = 2167.2

The value is 2167.2.

Answer:

5.04 \times (300 + 70 + 60) = 2167.2

4 · Reviewdoes it hold up?

Evaluate the original directly as a check: 1512 + 352.8 + 302.4 = 2167.2. The rewrite kept the value.

Another way: Factoring out 504 instead works too, but leaves fractions of a hundredth inside the brackets; taking the smallest of the three keeps every number whole.

Standardsmin grade 5

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate — Writing the factored form with parentheses and evaluating it.
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Relating the three decimals by powers of ten and finishing the multiplication.
3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Regrouping each product so the shared factor can come out.

💡Takeaway. Numbers that look different can share a factor once you move the decimal point's worth onto the other side.

Variant 12 hard

Rewrite the following as a simpler expression, then evaluate it.

$525 \times 4 + 52.5 \times 5 + 5.25 \times 55$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Three products are added. Their first factors -- 525, 52.5 and 5.25 -- are the same digits with the decimal point in different places. We must simplify before evaluating.

Givens

The expression is 525 x 4 + 52.5 x 5 + 5.25 x 55.
The three first factors share their digits.

Unknowns

A simpler equivalent expression, and its value.

Constraints

The rewrite must keep the same value, not just a similar shape.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #5 Look for a Pattern#7 Identify Subproblems

Nothing here is a common factor as written. But every first factor is 5.25 times a power of ten, so moving that power onto the other side of each product makes 5.25 common to all three.

3 · Execute4 carry out the plan

1Rewrite each decimal as a multiple of 5.25

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

525 is 5.25 times 100, 52.5 is 5.25 times 10, and 5.25 is 5.25 itself.

525 = 5.25 \times 100,\quad 52.5 = 5.25 \times 10

The same digits, three places apart.

2Turn each product into 5.25 times a number

#7 Identify Subproblems 3.OA.B.5

Multiplication can be regrouped, so the power of ten joins the other factor.

5.25 \times 400 + 5.25 \times 50 + 5.25 \times 55

Now 5.25 appears in every term.

3Factor out 5.25 with parentheses

#15 Organize Information in More Ways 5.OA.A.1

The distributive law runs backwards here: a common factor comes out and the rest goes inside brackets.

5.25 \times (400 + 50 + 55)

Three multiplications have become one.

4Add inside the parentheses, then multiply

#15 Organize Information in More Ways 5.NBT.B.7

The bracket is plain whole-number addition, and the final multiplication is by a round number.

5.25 \times 505 = 2651.25

The value is 2651.25.

Answer:

5.25 \times (400 + 50 + 55) = 2651.25

4 · Reviewdoes it hold up?

Evaluate the original directly as a check: 2100 + 262.5 + 288.75 = 2651.25. The rewrite kept the value.

Another way: Factoring out 525 instead works too, but leaves fractions of a hundredth inside the brackets; taking the smallest of the three keeps every number whole.

Standardsmin grade 5

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate — Writing the factored form with parentheses and evaluating it.
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Relating the three decimals by powers of ten and finishing the multiplication.
3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Regrouping each product so the shared factor can come out.

💡Takeaway. Numbers that look different can share a factor once you move the decimal point's worth onto the other side.

Factor out a common factor with parentheses · 12 practice problems

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