Practice · Find the amount for one unit, then scale · Sensim Math

Variant 1 easy answer:

19\frac{1}{2}

kg

An iron bar of uniform thickness weighs $4\frac{1}{3}\ \text{kg}$ for $\frac{2}{3}\ \text{m}$ . How many kilograms does $3\ \text{m}$ of this bar weigh?

Show solution

1 · Understandwhat's really being asked

A bar of even thickness weighs 4 and 1/3 kg for 2/3 of a metre. We want the weight of 3 metres.

Givens

2/3 m of the bar weighs 4 and 1/3 kg.
The bar is the same thickness all along.

Unknowns

The weight of 3 m of the bar.

Constraints

Weight is proportional to length, because the thickness never changes.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Jumping from 2/3 m to 3 m means multiplying by a ratio of fractions. Go through one metre instead: a division to get there, a multiplication to leave, and each step means something.

3 · Execute3 carry out the plan

1Write the weight as one fraction

#7 Identify Subproblems 6.NS.A.1

4 wholes are 12 3ths, plus 1 more.

4\frac{1}{3} = \frac{13}{3}

Easier to divide in this form.

2Find one metre's weight

#9 Solve an Easier Related Problem 7.RP.A.1

Divide by 2/3, which multiplies by 3/2.

\frac{13}{3} \div \frac{2}{3} = 6\frac{1}{2}

One metre weighs 6 and 1/2 kg.

3Scale up to 3 metres

#8 Analyze the Units 7.RP.A.1

Every metre weighs the same, so multiply.

6\frac{1}{2} \times 3 = 19\frac{1}{2}

3 metres weigh 19 and 1/2 kg.

Answer:

19\frac{1}{2}

kg

4 · Reviewdoes it hold up?

3 m is more than 2/3 m, and 19 and 1/2 kg is more than 4 and 1/3 kg -- and going back, $19\frac{1}{2} \div 3 \times \frac{2}{3} = \frac{13}{3}$ .

Another way: Scaling directly by the ratio of the two lengths, $3 \div \frac{2}{3}$ , reaches the same answer in one step -- and hides what one metre weighs, which is the number worth knowing.

Standardsmin grade 7

6.NS.A.1 Divide fractions by fractions — Dividing one fraction by another to get the rate.
7.RP.A.1 Compute unit rates associated with ratios of fractions — Reading the result as kilograms per metre and scaling it.

💡Takeaway. Find what one of something weighs and every other amount is one multiplication away.

Variant 2 easy answer:

4\frac{4}{5}

kg

An iron bar of uniform thickness weighs $1\frac{4}{5}\ \text{kg}$ for $\frac{3}{4}\ \text{m}$ . How many kilograms does $2\ \text{m}$ of this bar weigh?

Show solution

1 · Understandwhat's really being asked

A bar of even thickness weighs 1 and 4/5 kg for 3/4 of a metre. We want the weight of 2 metres.

Givens

3/4 m of the bar weighs 1 and 4/5 kg.
The bar is the same thickness all along.

Unknowns

The weight of 2 m of the bar.

Constraints

Weight is proportional to length, because the thickness never changes.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Jumping from 3/4 m to 2 m means multiplying by a ratio of fractions. Go through one metre instead: a division to get there, a multiplication to leave, and each step means something.

3 · Execute3 carry out the plan

1Write the weight as one fraction

#7 Identify Subproblems 6.NS.A.1

1 wholes are 5 5ths, plus 4 more.

1\frac{4}{5} = \frac{9}{5}

Easier to divide in this form.

2Find one metre's weight

#9 Solve an Easier Related Problem 7.RP.A.1

Divide by 3/4, which multiplies by 4/3.

\frac{9}{5} \div \frac{3}{4} = 2\frac{2}{5}

One metre weighs 2 and 2/5 kg.

3Scale up to 2 metres

#8 Analyze the Units 7.RP.A.1

Every metre weighs the same, so multiply.

2\frac{2}{5} \times 2 = 4\frac{4}{5}

2 metres weigh 4 and 4/5 kg.

Answer:

4\frac{4}{5}

kg

4 · Reviewdoes it hold up?

2 m is more than 3/4 m, and 4 and 4/5 kg is more than 1 and 4/5 kg -- and going back, $4\frac{4}{5} \div 2 \times \frac{3}{4} = \frac{9}{5}$ .

Another way: Scaling directly by the ratio of the two lengths, $2 \div \frac{3}{4}$ , reaches the same answer in one step -- and hides what one metre weighs, which is the number worth knowing.

Standardsmin grade 7

6.NS.A.1 Divide fractions by fractions — Dividing one fraction by another to get the rate.
7.RP.A.1 Compute unit rates associated with ratios of fractions — Reading the result as kilograms per metre and scaling it.

💡Takeaway. Find what one of something weighs and every other amount is one multiplication away.

Variant 3 easy answer:

18\frac{3}{4}

kg

An iron bar of uniform thickness weighs $2\frac{1}{2}\ \text{kg}$ for $\frac{2}{3}\ \text{m}$ . How many kilograms does $5\ \text{m}$ of this bar weigh?

Show solution

1 · Understandwhat's really being asked

A bar of even thickness weighs 2 and 1/2 kg for 2/3 of a metre. We want the weight of 5 metres.

Givens

2/3 m of the bar weighs 2 and 1/2 kg.
The bar is the same thickness all along.

Unknowns

The weight of 5 m of the bar.

Constraints

Weight is proportional to length, because the thickness never changes.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Jumping from 2/3 m to 5 m means multiplying by a ratio of fractions. Go through one metre instead: a division to get there, a multiplication to leave, and each step means something.

3 · Execute3 carry out the plan

1Write the weight as one fraction

#7 Identify Subproblems 6.NS.A.1

2 wholes are 4 2ths, plus 1 more.

2\frac{1}{2} = \frac{5}{2}

Easier to divide in this form.

2Find one metre's weight

#9 Solve an Easier Related Problem 7.RP.A.1

Divide by 2/3, which multiplies by 3/2.

\frac{5}{2} \div \frac{2}{3} = 3\frac{3}{4}

One metre weighs 3 and 3/4 kg.

3Scale up to 5 metres

#8 Analyze the Units 7.RP.A.1

Every metre weighs the same, so multiply.

3\frac{3}{4} \times 5 = 18\frac{3}{4}

5 metres weigh 18 and 3/4 kg.

Answer:

18\frac{3}{4}

kg

4 · Reviewdoes it hold up?

5 m is more than 2/3 m, and 18 and 3/4 kg is more than 2 and 1/2 kg -- and going back, $18\frac{3}{4} \div 5 \times \frac{2}{3} = \frac{5}{2}$ .

Another way: Scaling directly by the ratio of the two lengths, $5 \div \frac{2}{3}$ , reaches the same answer in one step -- and hides what one metre weighs, which is the number worth knowing.

Standardsmin grade 7

6.NS.A.1 Divide fractions by fractions — Dividing one fraction by another to get the rate.
7.RP.A.1 Compute unit rates associated with ratios of fractions — Reading the result as kilograms per metre and scaling it.

💡Takeaway. Find what one of something weighs and every other amount is one multiplication away.

Variant 4 easy answer:

28\frac{4}{5}

kg

An iron bar of uniform thickness weighs $5\frac{2}{5}\ \text{kg}$ for $\frac{3}{4}\ \text{m}$ . How many kilograms does $4\ \text{m}$ of this bar weigh?

Show solution

1 · Understandwhat's really being asked

A bar of even thickness weighs 5 and 2/5 kg for 3/4 of a metre. We want the weight of 4 metres.

Givens

3/4 m of the bar weighs 5 and 2/5 kg.
The bar is the same thickness all along.

Unknowns

The weight of 4 m of the bar.

Constraints

Weight is proportional to length, because the thickness never changes.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Jumping from 3/4 m to 4 m means multiplying by a ratio of fractions. Go through one metre instead: a division to get there, a multiplication to leave, and each step means something.

3 · Execute3 carry out the plan

1Write the weight as one fraction

#7 Identify Subproblems 6.NS.A.1

5 wholes are 25 5ths, plus 2 more.

5\frac{2}{5} = \frac{27}{5}

Easier to divide in this form.

2Find one metre's weight

#9 Solve an Easier Related Problem 7.RP.A.1

Divide by 3/4, which multiplies by 4/3.

\frac{27}{5} \div \frac{3}{4} = 7\frac{1}{5}

One metre weighs 7 and 1/5 kg.

3Scale up to 4 metres

#8 Analyze the Units 7.RP.A.1

Every metre weighs the same, so multiply.

7\frac{1}{5} \times 4 = 28\frac{4}{5}

4 metres weigh 28 and 4/5 kg.

Answer:

28\frac{4}{5}

kg

4 · Reviewdoes it hold up?

4 m is more than 3/4 m, and 28 and 4/5 kg is more than 5 and 2/5 kg -- and going back, $28\frac{4}{5} \div 4 \times \frac{3}{4} = \frac{27}{5}$ .

Another way: Scaling directly by the ratio of the two lengths, $4 \div \frac{3}{4}$ , reaches the same answer in one step -- and hides what one metre weighs, which is the number worth knowing.

Standardsmin grade 7

6.NS.A.1 Divide fractions by fractions — Dividing one fraction by another to get the rate.
7.RP.A.1 Compute unit rates associated with ratios of fractions — Reading the result as kilograms per metre and scaling it.

💡Takeaway. Find what one of something weighs and every other amount is one multiplication away.

Variant 5 medium answer:

12\frac{3}{5}

kg

An iron bar of uniform thickness weighs $5\frac{1}{4}\ \text{kg}$ for $\frac{5}{6}\ \text{m}$ . How many kilograms does $2\ \text{m}$ of this bar weigh?

Show solution

1 · Understandwhat's really being asked

A bar of even thickness weighs 5 and 1/4 kg for 5/6 of a metre. We want the weight of 2 metres.

Givens

5/6 m of the bar weighs 5 and 1/4 kg.
The bar is the same thickness all along.

Unknowns

The weight of 2 m of the bar.

Constraints

Weight is proportional to length, because the thickness never changes.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Jumping from 5/6 m to 2 m means multiplying by a ratio of fractions. Go through one metre instead: a division to get there, a multiplication to leave, and each step means something.

3 · Execute3 carry out the plan

1Write the weight as one fraction

#7 Identify Subproblems 6.NS.A.1

5 wholes are 20 4ths, plus 1 more.

5\frac{1}{4} = \frac{21}{4}

Easier to divide in this form.

2Find one metre's weight

#9 Solve an Easier Related Problem 7.RP.A.1

Divide by 5/6, which multiplies by 6/5.

\frac{21}{4} \div \frac{5}{6} = 6\frac{3}{10}

One metre weighs 6 and 3/10 kg.

3Scale up to 2 metres

#8 Analyze the Units 7.RP.A.1

Every metre weighs the same, so multiply.

6\frac{3}{10} \times 2 = 12\frac{3}{5}

2 metres weigh 12 and 3/5 kg.

Answer:

12\frac{3}{5}

kg

4 · Reviewdoes it hold up?

2 m is more than 5/6 m, and 12 and 3/5 kg is more than 5 and 1/4 kg -- and going back, $12\frac{3}{5} \div 2 \times \frac{5}{6} = \frac{21}{4}$ .

Another way: Scaling directly by the ratio of the two lengths, $2 \div \frac{5}{6}$ , reaches the same answer in one step -- and hides what one metre weighs, which is the number worth knowing.

Standardsmin grade 7

6.NS.A.1 Divide fractions by fractions — Dividing one fraction by another to get the rate.
7.RP.A.1 Compute unit rates associated with ratios of fractions — Reading the result as kilograms per metre and scaling it.

💡Takeaway. Find what one of something weighs and every other amount is one multiplication away.

Variant 6 medium answer:

37\frac{1}{2}

kg

An iron bar of uniform thickness weighs $1\frac{1}{4}\ \text{kg}$ for $\frac{1}{5}\ \text{m}$ . How many kilograms does $6\ \text{m}$ of this bar weigh?

Show solution

1 · Understandwhat's really being asked

A bar of even thickness weighs 1 and 1/4 kg for 1/5 of a metre. We want the weight of 6 metres.

Givens

1/5 m of the bar weighs 1 and 1/4 kg.
The bar is the same thickness all along.

Unknowns

The weight of 6 m of the bar.

Constraints

Weight is proportional to length, because the thickness never changes.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Jumping from 1/5 m to 6 m means multiplying by a ratio of fractions. Go through one metre instead: a division to get there, a multiplication to leave, and each step means something.

3 · Execute3 carry out the plan

1Write the weight as one fraction

#7 Identify Subproblems 6.NS.A.1

1 wholes are 4 4ths, plus 1 more.

1\frac{1}{4} = \frac{5}{4}

Easier to divide in this form.

2Find one metre's weight

#9 Solve an Easier Related Problem 7.RP.A.1

Divide by 1/5, which multiplies by 5/1.

\frac{5}{4} \div \frac{1}{5} = 6\frac{1}{4}

One metre weighs 6 and 1/4 kg.

3Scale up to 6 metres

#8 Analyze the Units 7.RP.A.1

Every metre weighs the same, so multiply.

6\frac{1}{4} \times 6 = 37\frac{1}{2}

6 metres weigh 37 and 1/2 kg.

Answer:

37\frac{1}{2}

kg

4 · Reviewdoes it hold up?

6 m is more than 1/5 m, and 37 and 1/2 kg is more than 1 and 1/4 kg -- and going back, $37\frac{1}{2} \div 6 \times \frac{1}{5} = \frac{5}{4}$ .

Another way: Scaling directly by the ratio of the two lengths, $6 \div \frac{1}{5}$ , reaches the same answer in one step -- and hides what one metre weighs, which is the number worth knowing.

Standardsmin grade 7

6.NS.A.1 Divide fractions by fractions — Dividing one fraction by another to get the rate.
7.RP.A.1 Compute unit rates associated with ratios of fractions — Reading the result as kilograms per metre and scaling it.

💡Takeaway. Find what one of something weighs and every other amount is one multiplication away.

Variant 7 medium answer:

16\frac{1}{3}

kg

An iron bar of uniform thickness weighs $3\frac{1}{2}\ \text{kg}$ for $\frac{3}{7}\ \text{m}$ . How many kilograms does $2\ \text{m}$ of this bar weigh?

Show solution

1 · Understandwhat's really being asked

A bar of even thickness weighs 3 and 1/2 kg for 3/7 of a metre. We want the weight of 2 metres.

Givens

3/7 m of the bar weighs 3 and 1/2 kg.
The bar is the same thickness all along.

Unknowns

The weight of 2 m of the bar.

Constraints

Weight is proportional to length, because the thickness never changes.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Jumping from 3/7 m to 2 m means multiplying by a ratio of fractions. Go through one metre instead: a division to get there, a multiplication to leave, and each step means something.

3 · Execute3 carry out the plan

1Write the weight as one fraction

#7 Identify Subproblems 6.NS.A.1

3 wholes are 6 2ths, plus 1 more.

3\frac{1}{2} = \frac{7}{2}

Easier to divide in this form.

2Find one metre's weight

#9 Solve an Easier Related Problem 7.RP.A.1

Divide by 3/7, which multiplies by 7/3.

\frac{7}{2} \div \frac{3}{7} = 8\frac{1}{6}

One metre weighs 8 and 1/6 kg.

3Scale up to 2 metres

#8 Analyze the Units 7.RP.A.1

Every metre weighs the same, so multiply.

8\frac{1}{6} \times 2 = 16\frac{1}{3}

2 metres weigh 16 and 1/3 kg.

Answer:

16\frac{1}{3}

kg

4 · Reviewdoes it hold up?

2 m is more than 3/7 m, and 16 and 1/3 kg is more than 3 and 1/2 kg -- and going back, $16\frac{1}{3} \div 2 \times \frac{3}{7} = \frac{7}{2}$ .

Another way: Scaling directly by the ratio of the two lengths, $2 \div \frac{3}{7}$ , reaches the same answer in one step -- and hides what one metre weighs, which is the number worth knowing.

Standardsmin grade 7

6.NS.A.1 Divide fractions by fractions — Dividing one fraction by another to get the rate.
7.RP.A.1 Compute unit rates associated with ratios of fractions — Reading the result as kilograms per metre and scaling it.

💡Takeaway. Find what one of something weighs and every other amount is one multiplication away.

Variant 8 hard answer:

12\frac{5}{6}

kg

An iron bar of uniform thickness weighs $5\frac{1}{2}\ \text{kg}$ for $\frac{6}{7}\ \text{m}$ . How many kilograms does $2\ \text{m}$ of this bar weigh?

Show solution

1 · Understandwhat's really being asked

A bar of even thickness weighs 5 and 1/2 kg for 6/7 of a metre. We want the weight of 2 metres.

Givens

6/7 m of the bar weighs 5 and 1/2 kg.
The bar is the same thickness all along.

Unknowns

The weight of 2 m of the bar.

Constraints

Weight is proportional to length, because the thickness never changes.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Jumping from 6/7 m to 2 m means multiplying by a ratio of fractions. Go through one metre instead: a division to get there, a multiplication to leave, and each step means something.

3 · Execute3 carry out the plan

1Write the weight as one fraction

#7 Identify Subproblems 6.NS.A.1

5 wholes are 10 2ths, plus 1 more.

5\frac{1}{2} = \frac{11}{2}

Easier to divide in this form.

2Find one metre's weight

#9 Solve an Easier Related Problem 7.RP.A.1

Divide by 6/7, which multiplies by 7/6.

\frac{11}{2} \div \frac{6}{7} = 6\frac{5}{12}

One metre weighs 6 and 5/12 kg.

3Scale up to 2 metres

#8 Analyze the Units 7.RP.A.1

Every metre weighs the same, so multiply.

6\frac{5}{12} \times 2 = 12\frac{5}{6}

2 metres weigh 12 and 5/6 kg.

Answer:

12\frac{5}{6}

kg

4 · Reviewdoes it hold up?

2 m is more than 6/7 m, and 12 and 5/6 kg is more than 5 and 1/2 kg -- and going back, $12\frac{5}{6} \div 2 \times \frac{6}{7} = \frac{11}{2}$ .

Another way: Scaling directly by the ratio of the two lengths, $2 \div \frac{6}{7}$ , reaches the same answer in one step -- and hides what one metre weighs, which is the number worth knowing.

Standardsmin grade 7

6.NS.A.1 Divide fractions by fractions — Dividing one fraction by another to get the rate.
7.RP.A.1 Compute unit rates associated with ratios of fractions — Reading the result as kilograms per metre and scaling it.

💡Takeaway. Find what one of something weighs and every other amount is one multiplication away.

Variant 9 medium answer:

44\frac{5}{8}

kg

An iron bar of uniform thickness weighs $4\frac{1}{4}\ \text{kg}$ for $\frac{2}{7}\ \text{m}$ . How many kilograms does $3\ \text{m}$ of this bar weigh?

Show solution

1 · Understandwhat's really being asked

A bar of even thickness weighs 4 and 1/4 kg for 2/7 of a metre. We want the weight of 3 metres.

Givens

2/7 m of the bar weighs 4 and 1/4 kg.
The bar is the same thickness all along.

Unknowns

The weight of 3 m of the bar.

Constraints

Weight is proportional to length, because the thickness never changes.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Jumping from 2/7 m to 3 m means multiplying by a ratio of fractions. Go through one metre instead: a division to get there, a multiplication to leave, and each step means something.

3 · Execute3 carry out the plan

1Write the weight as one fraction

#7 Identify Subproblems 6.NS.A.1

4 wholes are 16 4ths, plus 1 more.

4\frac{1}{4} = \frac{17}{4}

Easier to divide in this form.

2Find one metre's weight

#9 Solve an Easier Related Problem 7.RP.A.1

Divide by 2/7, which multiplies by 7/2.

\frac{17}{4} \div \frac{2}{7} = 14\frac{7}{8}

One metre weighs 14 and 7/8 kg.

3Scale up to 3 metres

#8 Analyze the Units 7.RP.A.1

Every metre weighs the same, so multiply.

14\frac{7}{8} \times 3 = 44\frac{5}{8}

3 metres weigh 44 and 5/8 kg.

Answer:

44\frac{5}{8}

kg

4 · Reviewdoes it hold up?

3 m is more than 2/7 m, and 44 and 5/8 kg is more than 4 and 1/4 kg -- and going back, $44\frac{5}{8} \div 3 \times \frac{2}{7} = \frac{17}{4}$ .

Another way: Scaling directly by the ratio of the two lengths, $3 \div \frac{2}{7}$ , reaches the same answer in one step -- and hides what one metre weighs, which is the number worth knowing.

Standardsmin grade 7

6.NS.A.1 Divide fractions by fractions — Dividing one fraction by another to get the rate.
7.RP.A.1 Compute unit rates associated with ratios of fractions — Reading the result as kilograms per metre and scaling it.

💡Takeaway. Find what one of something weighs and every other amount is one multiplication away.

Variant 10 hard answer:

31\frac{1}{3}

kg

An iron bar of uniform thickness weighs $5\frac{7}{8}\ \text{kg}$ for $\frac{3}{8}\ \text{m}$ . How many kilograms does $2\ \text{m}$ of this bar weigh?

Show solution

1 · Understandwhat's really being asked

A bar of even thickness weighs 5 and 7/8 kg for 3/8 of a metre. We want the weight of 2 metres.

Givens

3/8 m of the bar weighs 5 and 7/8 kg.
The bar is the same thickness all along.

Unknowns

The weight of 2 m of the bar.

Constraints

Weight is proportional to length, because the thickness never changes.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Jumping from 3/8 m to 2 m means multiplying by a ratio of fractions. Go through one metre instead: a division to get there, a multiplication to leave, and each step means something.

3 · Execute3 carry out the plan

1Write the weight as one fraction

#7 Identify Subproblems 6.NS.A.1

5 wholes are 40 8ths, plus 7 more.

5\frac{7}{8} = \frac{47}{8}

Easier to divide in this form.

2Find one metre's weight

#9 Solve an Easier Related Problem 7.RP.A.1

Divide by 3/8, which multiplies by 8/3.

\frac{47}{8} \div \frac{3}{8} = 15\frac{2}{3}

One metre weighs 15 and 2/3 kg.

3Scale up to 2 metres

#8 Analyze the Units 7.RP.A.1

Every metre weighs the same, so multiply.

15\frac{2}{3} \times 2 = 31\frac{1}{3}

2 metres weigh 31 and 1/3 kg.

Answer:

31\frac{1}{3}

kg

4 · Reviewdoes it hold up?

2 m is more than 3/8 m, and 31 and 1/3 kg is more than 5 and 7/8 kg -- and going back, $31\frac{1}{3} \div 2 \times \frac{3}{8} = \frac{47}{8}$ .

Another way: Scaling directly by the ratio of the two lengths, $2 \div \frac{3}{8}$ , reaches the same answer in one step -- and hides what one metre weighs, which is the number worth knowing.

Standardsmin grade 7

6.NS.A.1 Divide fractions by fractions — Dividing one fraction by another to get the rate.
7.RP.A.1 Compute unit rates associated with ratios of fractions — Reading the result as kilograms per metre and scaling it.

💡Takeaway. Find what one of something weighs and every other amount is one multiplication away.

Variant 11 hard answer:

21\frac{1}{7}

kg

An iron bar of uniform thickness weighs $3\frac{7}{10}\ \text{kg}$ for $\frac{7}{10}\ \text{m}$ . How many kilograms does $4\ \text{m}$ of this bar weigh?

Show solution

1 · Understandwhat's really being asked

A bar of even thickness weighs 3 and 7/10 kg for 7/10 of a metre. We want the weight of 4 metres.

Givens

7/10 m of the bar weighs 3 and 7/10 kg.
The bar is the same thickness all along.

Unknowns

The weight of 4 m of the bar.

Constraints

Weight is proportional to length, because the thickness never changes.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Jumping from 7/10 m to 4 m means multiplying by a ratio of fractions. Go through one metre instead: a division to get there, a multiplication to leave, and each step means something.

3 · Execute3 carry out the plan

1Write the weight as one fraction

#7 Identify Subproblems 6.NS.A.1

3 wholes are 30 10ths, plus 7 more.

3\frac{7}{10} = \frac{37}{10}

Easier to divide in this form.

2Find one metre's weight

#9 Solve an Easier Related Problem 7.RP.A.1

Divide by 7/10, which multiplies by 10/7.

\frac{37}{10} \div \frac{7}{10} = 5\frac{2}{7}

One metre weighs 5 and 2/7 kg.

3Scale up to 4 metres

#8 Analyze the Units 7.RP.A.1

Every metre weighs the same, so multiply.

5\frac{2}{7} \times 4 = 21\frac{1}{7}

4 metres weigh 21 and 1/7 kg.

Answer:

21\frac{1}{7}

kg

4 · Reviewdoes it hold up?

4 m is more than 7/10 m, and 21 and 1/7 kg is more than 3 and 7/10 kg -- and going back, $21\frac{1}{7} \div 4 \times \frac{7}{10} = \frac{37}{10}$ .

Another way: Scaling directly by the ratio of the two lengths, $4 \div \frac{7}{10}$ , reaches the same answer in one step -- and hides what one metre weighs, which is the number worth knowing.

Standardsmin grade 7

6.NS.A.1 Divide fractions by fractions — Dividing one fraction by another to get the rate.
7.RP.A.1 Compute unit rates associated with ratios of fractions — Reading the result as kilograms per metre and scaling it.

💡Takeaway. Find what one of something weighs and every other amount is one multiplication away.

Variant 12 hard answer:

26\frac{2}{9}

kg

An iron bar of uniform thickness weighs $5\frac{9}{10}\ \text{kg}$ for $\frac{9}{10}\ \text{m}$ . How many kilograms does $4\ \text{m}$ of this bar weigh?

Show solution

1 · Understandwhat's really being asked

A bar of even thickness weighs 5 and 9/10 kg for 9/10 of a metre. We want the weight of 4 metres.

Givens

9/10 m of the bar weighs 5 and 9/10 kg.
The bar is the same thickness all along.

Unknowns

The weight of 4 m of the bar.

Constraints

Weight is proportional to length, because the thickness never changes.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Jumping from 9/10 m to 4 m means multiplying by a ratio of fractions. Go through one metre instead: a division to get there, a multiplication to leave, and each step means something.

3 · Execute3 carry out the plan

1Write the weight as one fraction

#7 Identify Subproblems 6.NS.A.1

5 wholes are 50 10ths, plus 9 more.

5\frac{9}{10} = \frac{59}{10}

Easier to divide in this form.

2Find one metre's weight

#9 Solve an Easier Related Problem 7.RP.A.1

Divide by 9/10, which multiplies by 10/9.

\frac{59}{10} \div \frac{9}{10} = 6\frac{5}{9}

One metre weighs 6 and 5/9 kg.

3Scale up to 4 metres

#8 Analyze the Units 7.RP.A.1

Every metre weighs the same, so multiply.

6\frac{5}{9} \times 4 = 26\frac{2}{9}

4 metres weigh 26 and 2/9 kg.

Answer:

26\frac{2}{9}

kg

4 · Reviewdoes it hold up?

4 m is more than 9/10 m, and 26 and 2/9 kg is more than 5 and 9/10 kg -- and going back, $26\frac{2}{9} \div 4 \times \frac{9}{10} = \frac{59}{10}$ .

Another way: Scaling directly by the ratio of the two lengths, $4 \div \frac{9}{10}$ , reaches the same answer in one step -- and hides what one metre weighs, which is the number worth knowing.

Standardsmin grade 7

6.NS.A.1 Divide fractions by fractions — Dividing one fraction by another to get the rate.
7.RP.A.1 Compute unit rates associated with ratios of fractions — Reading the result as kilograms per metre and scaling it.

💡Takeaway. Find what one of something weighs and every other amount is one multiplication away.

Find the amount for one unit, then scale · 12 practice problems

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