← Solve missing data from ratio or sum clues · Solve a Table or Graph Step by Step from Clues

Solve missing data from ratio or sum clues · 8 practice problems

3.MD.B.33.NF.A.13.NBT.A.2

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Generated variants — 8

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

Variant 1 easy

The amount for each item was surveyed and shown in a pictograph. Find the amount for the Vera item.

Amount by Item (pictograph)

Item	Amount
Lila
Rosa	(6 large pictures, 0 small pictures)
Vera
Nina

In the pictograph, each large picture stands for $100\ \text{units}$ and each small picture stands for $10\ \text{units}$ .

Conditions

The Lila item has $40$ fewer than the Rosa item.
The Nina item has $\dfrac{3}{5}$ as many as the Lila item.
The Vera item has $90$ more than the Nina item.

Your answer

Show solution

1 · Understandwhat's really being asked

A pictograph gives Rosa as 6 large picture symbols (100 each) and 0 small (10 each). Three clues link Lila, Nina, and Vera to one another. We find Vera's amount.

Givens

Rosa (from figure) = 6 large + 0 small symbols; 1 large = 100, 1 small = 10 units.
Lila = Rosa - 40.
Nina = $\frac{3}{5}$ of Lila.
Vera = Nina + 90.

Unknowns

The amount for the Vera item.

Constraints

Amounts are whole numbers.
For Nina to be a whole number, Lila must be a multiple of 5.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems

Each clue depends on the previous one, so we solve in order (Rosa -> Lila -> Nina -> Vera), each a small subproblem, chaining toward Vera.

3 · Execute4 carry out the plan

1Read Rosa from the pictograph

#7 Identify Subproblems 3.MD.B.3

Rosa shows 6 large symbols and 0 small ones. Each large is 100 and each small is 10: 6 x 100 + 0 x 10 = 600.

6 \times 100 + 0 \times 10 = 600

Turning the scaled symbols into a count is the core skill of a scaled pictograph.

2Find Lila

#11 Work Backwards 3.NBT.A.2

Lila has 40 fewer than Rosa: 600 - 40 = 560. (This is a multiple of 5, so the next fraction step gives a whole number.)

600 - 40 = 560

Subtracting within the thousands is Grade 3 place-value work.

3Find Nina ( $\frac{3}{5}$ of Lila)

#7 Identify Subproblems 3.NF.A.1

Nina is

\frac{3}{5}

of Lila. Splitting 560 into 5 equal parts gives 560 / 5 = 112, and taking 3 of them gives Nina = 3 x 112 = 336.

\tfrac{3}{5}\times 560 = 3 \times 112 = 336

Understanding

\frac{3}{5}

as 3 of the 5 equal parts of a whole is the Grade 3 meaning of a fraction.

4Find Vera

#11 Work Backwards 3.NBT.A.2

Vera has 90 more than Nina: 336 + 90 = 426.

336 + 90 = 426

Adding 90 to 336 is a clean within-1000 addition.

Answer: 426 units

4 · Reviewdoes it hold up?

Vera (426) follows from the chain: the $\frac{3}{5}$ clue forces Lila to be a multiple of 5; 560 = 5 x 112, so Nina = 3 x 112 = 336 and Vera = 336 + 90 = 426, all whole numbers.

Another way: Work the fraction as a subproblem first: one part of 560 is 112, so $\frac{3}{5}$ is 3 x 112 = 336, then add 90 to reach 426 - same answer.

Standardsmin grade 3

3.MD.B.3 Draw and interpret scaled picture graphs and bar graphs — Reading Rosa's value from the large and small symbols.
3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Interpreting $\frac{3}{5}$ of Lila as 3 of its 5 equal parts.
3.NBT.A.2 Fluently add and subtract within 1000 — Subtracting 40 and adding 90 along the chain of clues.

💡Takeaway. Solve the clues one after another - read the graph, then 3 of the 5 equal parts, then add 90!

Variant 2 easy

The amount for each item was surveyed and shown in a pictograph. Find the amount for the Lead item.

Amount by Item (pictograph)

Item	Amount
Iron
Tin	(7 large pictures, 2 small pictures)
Lead
Zinc

In the pictograph, each large picture stands for $100\ \text{kg}$ and each small picture stands for $10\ \text{kg}$ .

Conditions

The Iron item has $40$ fewer than the Tin item.
The Zinc item has $\dfrac{4}{5}$ as many as the Iron item.
The Lead item has $80$ more than the Zinc item.

Your answer

Show solution

1 · Understandwhat's really being asked

A pictograph gives Tin as 7 large picture symbols (100 each) and 2 small (10 each). Three clues link Iron, Zinc, and Lead to one another. We find Lead's amount.

Givens

Tin (from figure) = 7 large + 2 small symbols; 1 large = 100, 1 small = 10 kg.
Iron = Tin - 40.
Zinc = $\frac{4}{5}$ of Iron.
Lead = Zinc + 80.

Unknowns

The amount for the Lead item.

Constraints

Amounts are whole numbers.
For Zinc to be a whole number, Iron must be a multiple of 5.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems

Each clue depends on the previous one, so we solve in order (Tin -> Iron -> Zinc -> Lead), each a small subproblem, chaining toward Lead.

3 · Execute4 carry out the plan

1Read Tin from the pictograph

#7 Identify Subproblems 3.MD.B.3

Tin shows 7 large symbols and 2 small ones. Each large is 100 and each small is 10: 7 x 100 + 2 x 10 = 720.

7 \times 100 + 2 \times 10 = 720

Turning the scaled symbols into a count is the core skill of a scaled pictograph.

2Find Iron

#11 Work Backwards 3.NBT.A.2

Iron has 40 fewer than Tin: 720 - 40 = 680. (This is a multiple of 5, so the next fraction step gives a whole number.)

720 - 40 = 680

Subtracting within the thousands is Grade 3 place-value work.

3Find Zinc ( $\frac{4}{5}$ of Iron)

#7 Identify Subproblems 3.NF.A.1

Zinc is

\frac{4}{5}

of Iron. Splitting 680 into 5 equal parts gives 680 / 5 = 136, and taking 4 of them gives Zinc = 4 x 136 = 544.

\tfrac{4}{5}\times 680 = 4 \times 136 = 544

Understanding

\frac{4}{5}

as 4 of the 5 equal parts of a whole is the Grade 3 meaning of a fraction.

4Find Lead

#11 Work Backwards 3.NBT.A.2

Lead has 80 more than Zinc: 544 + 80 = 624.

544 + 80 = 624

Adding 80 to 544 is a clean within-1000 addition.

Answer: 624 kg

4 · Reviewdoes it hold up?

Lead (624) follows from the chain: the $\frac{4}{5}$ clue forces Iron to be a multiple of 5; 680 = 5 x 136, so Zinc = 4 x 136 = 544 and Lead = 544 + 80 = 624, all whole numbers.

Another way: Work the fraction as a subproblem first: one part of 680 is 136, so $\frac{4}{5}$ is 4 x 136 = 544, then add 80 to reach 624 - same answer.

Standardsmin grade 3

3.MD.B.3 Draw and interpret scaled picture graphs and bar graphs — Reading Tin's value from the large and small symbols.
3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Interpreting $\frac{4}{5}$ of Iron as 4 of its 5 equal parts.
3.NBT.A.2 Fluently add and subtract within 1000 — Subtracting 40 and adding 80 along the chain of clues.

💡Takeaway. Solve the clues one after another - read the graph, then 4 of the 5 equal parts, then add 80!

Variant 3 medium

The amount for each item was surveyed and shown in a pictograph. Find the amount for the Fir item.

Amount by Item (pictograph)

Item	Amount
Ash
Elm	(8 large pictures, 0 small pictures)
Fir
Yew

In the pictograph, each large picture stands for $100\ \text{trees}$ and each small picture stands for $10\ \text{trees}$ .

Conditions

The Ash item has $100$ fewer than the Elm item.
The Yew item has $\dfrac{3}{4}$ as many as the Ash item.
The Fir item has $60$ more than the Yew item.

Your answer

Show solution

1 · Understandwhat's really being asked

A pictograph gives Elm as 8 large picture symbols (100 each) and 0 small (10 each). Three clues link Ash, Yew, and Fir to one another. We find Fir's amount.

Givens

Elm (from figure) = 8 large + 0 small symbols; 1 large = 100, 1 small = 10 trees.
Ash = Elm - 100.
Yew = $\frac{3}{4}$ of Ash.
Fir = Yew + 60.

Unknowns

The amount for the Fir item.

Constraints

Amounts are whole numbers.
For Yew to be a whole number, Ash must be a multiple of 4.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems

Each clue depends on the previous one, so we solve in order (Elm -> Ash -> Yew -> Fir), each a small subproblem, chaining toward Fir.

3 · Execute4 carry out the plan

1Read Elm from the pictograph

#7 Identify Subproblems 3.MD.B.3

Elm shows 8 large symbols and 0 small ones. Each large is 100 and each small is 10: 8 x 100 + 0 x 10 = 800.

8 \times 100 + 0 \times 10 = 800

Turning the scaled symbols into a count is the core skill of a scaled pictograph.

2Find Ash

#11 Work Backwards 3.NBT.A.2

Ash has 100 fewer than Elm: 800 - 100 = 700. (This is a multiple of 4, so the next fraction step gives a whole number.)

800 - 100 = 700

Subtracting within the thousands is Grade 3 place-value work.

3Find Yew ( $\frac{3}{4}$ of Ash)

#7 Identify Subproblems 3.NF.A.1

Yew is

\frac{3}{4}

of Ash. Splitting 700 into 4 equal parts gives 700 / 4 = 175, and taking 3 of them gives Yew = 3 x 175 = 525.

\tfrac{3}{4}\times 700 = 3 \times 175 = 525

Understanding

\frac{3}{4}

as 3 of the 4 equal parts of a whole is the Grade 3 meaning of a fraction.

4Find Fir

#11 Work Backwards 3.NBT.A.2

Fir has 60 more than Yew: 525 + 60 = 585.

525 + 60 = 585

Adding 60 to 525 is a clean within-1000 addition.

Answer: 585 trees

4 · Reviewdoes it hold up?

Fir (585) follows from the chain: the $\frac{3}{4}$ clue forces Ash to be a multiple of 4; 700 = 4 x 175, so Yew = 3 x 175 = 525 and Fir = 525 + 60 = 585, all whole numbers.

Another way: Work the fraction as a subproblem first: one part of 700 is 175, so $\frac{3}{4}$ is 3 x 175 = 525, then add 60 to reach 585 - same answer.

Standardsmin grade 3

3.MD.B.3 Draw and interpret scaled picture graphs and bar graphs — Reading Elm's value from the large and small symbols.
3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Interpreting $\frac{3}{4}$ of Ash as 3 of its 4 equal parts.
3.NBT.A.2 Fluently add and subtract within 1000 — Subtracting 100 and adding 60 along the chain of clues.

💡Takeaway. Solve the clues one after another - read the graph, then 3 of the 4 equal parts, then add 60!

Variant 4 medium

The amount for each item was surveyed and shown in a pictograph. Find the amount for the Elk item.

Amount by Item (pictograph)

Item	Amount
Fox
Owl	(3 large pictures, 6 small pictures)
Elk
Doe

In the pictograph, each large picture stands for $100\ \text{animals}$ and each small picture stands for $10\ \text{animals}$ .

Conditions

The Fox item has $90$ fewer than the Owl item.
The Doe item has $\dfrac{5}{6}$ as many as the Fox item.
The Elk item has $100$ more than the Doe item.

Your answer

Show solution

1 · Understandwhat's really being asked

A pictograph gives Owl as 3 large picture symbols (100 each) and 6 small (10 each). Three clues link Fox, Doe, and Elk to one another. We find Elk's amount.

Givens

Owl (from figure) = 3 large + 6 small symbols; 1 large = 100, 1 small = 10 animals.
Fox = Owl - 90.
Doe = $\frac{5}{6}$ of Fox.
Elk = Doe + 100.

Unknowns

The amount for the Elk item.

Constraints

Amounts are whole numbers.
For Doe to be a whole number, Fox must be a multiple of 6.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems

Each clue depends on the previous one, so we solve in order (Owl -> Fox -> Doe -> Elk), each a small subproblem, chaining toward Elk.

3 · Execute4 carry out the plan

1Read Owl from the pictograph

#7 Identify Subproblems 3.MD.B.3

Owl shows 3 large symbols and 6 small ones. Each large is 100 and each small is 10: 3 x 100 + 6 x 10 = 360.

3 \times 100 + 6 \times 10 = 360

Turning the scaled symbols into a count is the core skill of a scaled pictograph.

2Find Fox

#11 Work Backwards 3.NBT.A.2

Fox has 90 fewer than Owl: 360 - 90 = 270. (This is a multiple of 6, so the next fraction step gives a whole number.)

360 - 90 = 270

Subtracting within the thousands is Grade 3 place-value work.

3Find Doe ( $\frac{5}{6}$ of Fox)

#7 Identify Subproblems 3.NF.A.1

Doe is

\frac{5}{6}

of Fox. Splitting 270 into 6 equal parts gives 270 / 6 = 45, and taking 5 of them gives Doe = 5 x 45 = 225.

\tfrac{5}{6}\times 270 = 5 \times 45 = 225

Understanding

\frac{5}{6}

as 5 of the 6 equal parts of a whole is the Grade 3 meaning of a fraction.

4Find Elk

#11 Work Backwards 3.NBT.A.2

Elk has 100 more than Doe: 225 + 100 = 325.

225 + 100 = 325

Adding 100 to 225 is a clean within-1000 addition.

Answer: 325 animals

4 · Reviewdoes it hold up?

Elk (325) follows from the chain: the $\frac{5}{6}$ clue forces Fox to be a multiple of 6; 270 = 6 x 45, so Doe = 5 x 45 = 225 and Elk = 225 + 100 = 325, all whole numbers.

Another way: Work the fraction as a subproblem first: one part of 270 is 45, so $\frac{5}{6}$ is 5 x 45 = 225, then add 100 to reach 325 - same answer.

Standardsmin grade 3

3.MD.B.3 Draw and interpret scaled picture graphs and bar graphs — Reading Owl's value from the large and small symbols.
3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Interpreting $\frac{5}{6}$ of Fox as 5 of its 6 equal parts.
3.NBT.A.2 Fluently add and subtract within 1000 — Subtracting 90 and adding 100 along the chain of clues.

💡Takeaway. Solve the clues one after another - read the graph, then 5 of the 6 equal parts, then add 100!

Variant 5 medium

The amount for each item was surveyed and shown in a pictograph. Find the amount for the Lou item.

Amount by Item (pictograph)

Item	Amount
Pat
Sam	(5 large pictures, 5 small pictures)
Lou
Kim

In the pictograph, each large picture stands for $100\ \text{books}$ and each small picture stands for $10\ \text{books}$ .

Conditions

The Pat item has $50$ fewer than the Sam item.
The Kim item has $\dfrac{4}{5}$ as many as the Pat item.
The Lou item has $110$ more than the Kim item.

Your answer

Show solution

1 · Understandwhat's really being asked

A pictograph gives Sam as 5 large picture symbols (100 each) and 5 small (10 each). Three clues link Pat, Kim, and Lou to one another. We find Lou's amount.

Givens

Sam (from figure) = 5 large + 5 small symbols; 1 large = 100, 1 small = 10 books.
Pat = Sam - 50.
Kim = $\frac{4}{5}$ of Pat.
Lou = Kim + 110.

Unknowns

The amount for the Lou item.

Constraints

Amounts are whole numbers.
For Kim to be a whole number, Pat must be a multiple of 5.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems

Each clue depends on the previous one, so we solve in order (Sam -> Pat -> Kim -> Lou), each a small subproblem, chaining toward Lou.

3 · Execute4 carry out the plan

1Read Sam from the pictograph

#7 Identify Subproblems 3.MD.B.3

Sam shows 5 large symbols and 5 small ones. Each large is 100 and each small is 10: 5 x 100 + 5 x 10 = 550.

5 \times 100 + 5 \times 10 = 550

Turning the scaled symbols into a count is the core skill of a scaled pictograph.

2Find Pat

#11 Work Backwards 3.NBT.A.2

Pat has 50 fewer than Sam: 550 - 50 = 500. (This is a multiple of 5, so the next fraction step gives a whole number.)

550 - 50 = 500

Subtracting within the thousands is Grade 3 place-value work.

3Find Kim ( $\frac{4}{5}$ of Pat)

#7 Identify Subproblems 3.NF.A.1

Kim is

\frac{4}{5}

of Pat. Splitting 500 into 5 equal parts gives 500 / 5 = 100, and taking 4 of them gives Kim = 4 x 100 = 400.

\tfrac{4}{5}\times 500 = 4 \times 100 = 400

Understanding

\frac{4}{5}

as 4 of the 5 equal parts of a whole is the Grade 3 meaning of a fraction.

4Find Lou

#11 Work Backwards 3.NBT.A.2

Lou has 110 more than Kim: 400 + 110 = 510.

400 + 110 = 510

Adding 110 to 400 is a clean within-1000 addition.

Answer: 510 books

4 · Reviewdoes it hold up?

Lou (510) follows from the chain: the $\frac{4}{5}$ clue forces Pat to be a multiple of 5; 500 = 5 x 100, so Kim = 4 x 100 = 400 and Lou = 400 + 110 = 510, all whole numbers.

Another way: Work the fraction as a subproblem first: one part of 500 is 100, so $\frac{4}{5}$ is 4 x 100 = 400, then add 110 to reach 510 - same answer.

Standardsmin grade 3

3.MD.B.3 Draw and interpret scaled picture graphs and bar graphs — Reading Sam's value from the large and small symbols.
3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Interpreting $\frac{4}{5}$ of Pat as 4 of its 5 equal parts.
3.NBT.A.2 Fluently add and subtract within 1000 — Subtracting 50 and adding 110 along the chain of clues.

💡Takeaway. Solve the clues one after another - read the graph, then 4 of the 5 equal parts, then add 110!

Variant 6 medium

The amount for each item was surveyed and shown in a pictograph. Find the amount for the Mugunghwa item.

Amount by Item (pictograph)

Item	Amount
Dalbit
Chowon	(5 large pictures, 3 small pictures)
Mugunghwa
Taeyang

In the pictograph, each large picture stands for $100\ \text{households}$ and each small picture stands for $10\ \text{households}$ .

Conditions

The Dalbit item has $40$ fewer than the Chowon item.
The Taeyang item has $\dfrac{5}{7}$ as many as the Dalbit item.
The Mugunghwa item has $120$ more than the Taeyang item.

Your answer

Show solution

1 · Understandwhat's really being asked

A pictograph gives Chowon as 5 large picture symbols (100 each) and 3 small (10 each). Three clues link Dalbit, Taeyang, and Mugunghwa to one another. We find Mugunghwa's amount.

Givens

Chowon (from figure) = 5 large + 3 small symbols; 1 large = 100, 1 small = 10 households.
Dalbit = Chowon - 40.
Taeyang = $\frac{5}{7}$ of Dalbit.
Mugunghwa = Taeyang + 120.

Unknowns

The amount for the Mugunghwa item.

Constraints

Amounts are whole numbers.
For Taeyang to be a whole number, Dalbit must be a multiple of 7.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems

Each clue depends on the previous one, so we solve in order (Chowon -> Dalbit -> Taeyang -> Mugunghwa), each a small subproblem, chaining toward Mugunghwa.

3 · Execute4 carry out the plan

1Read Chowon from the pictograph

#7 Identify Subproblems 3.MD.B.3

Chowon shows 5 large symbols and 3 small ones. Each large is 100 and each small is 10: 5 x 100 + 3 x 10 = 530.

5 \times 100 + 3 \times 10 = 530

Turning the scaled symbols into a count is the core skill of a scaled pictograph.

2Find Dalbit

#11 Work Backwards 3.NBT.A.2

Dalbit has 40 fewer than Chowon: 530 - 40 = 490. (This is a multiple of 7, so the next fraction step gives a whole number.)

530 - 40 = 490

Subtracting within the thousands is Grade 3 place-value work.

3Find Taeyang ( $\frac{5}{7}$ of Dalbit)

#7 Identify Subproblems 3.NF.A.1

Taeyang is

\frac{5}{7}

of Dalbit. Splitting 490 into 7 equal parts gives 490 / 7 = 70, and taking 5 of them gives Taeyang = 5 x 70 = 350.

\tfrac{5}{7}\times 490 = 5 \times 70 = 350

Understanding

\frac{5}{7}

as 5 of the 7 equal parts of a whole is the Grade 3 meaning of a fraction.

4Find Mugunghwa

#11 Work Backwards 3.NBT.A.2

Mugunghwa has 120 more than Taeyang: 350 + 120 = 470.

350 + 120 = 470

Adding 120 to 350 is a clean within-1000 addition.

Answer: 470 households

4 · Reviewdoes it hold up?

Mugunghwa (470) follows from the chain: the $\frac{5}{7}$ clue forces Dalbit to be a multiple of 7; 490 = 7 x 70, so Taeyang = 5 x 70 = 350 and Mugunghwa = 350 + 120 = 470, all whole numbers.

Another way: Work the fraction as a subproblem first: one part of 490 is 70, so $\frac{5}{7}$ is 5 x 70 = 350, then add 120 to reach 470 - same answer.

Standardsmin grade 3

3.MD.B.3 Draw and interpret scaled picture graphs and bar graphs — Reading Chowon's value from the large and small symbols.
3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Interpreting $\frac{5}{7}$ of Dalbit as 5 of its 7 equal parts.
3.NBT.A.2 Fluently add and subtract within 1000 — Subtracting 40 and adding 120 along the chain of clues.

💡Takeaway. Solve the clues one after another - read the graph, then 5 of the 7 equal parts, then add 120!

Variant 7 hard

The amount for each item was surveyed and shown in a pictograph. Find the amount for the Lime item.

Amount by Item (pictograph)

Item	Amount
Cyan
Teal	(6 large pictures, 4 small pictures)
Lime
Plum

In the pictograph, each large picture stands for $100\ \text{tickets}$ and each small picture stands for $10\ \text{tickets}$ .

Conditions

The Cyan item has $40$ fewer than the Teal item.
The Plum item has $\dfrac{2}{3}$ as many as the Cyan item.
The Lime item has $130$ more than the Plum item.

Your answer

Show solution

1 · Understandwhat's really being asked

A pictograph gives Teal as 6 large picture symbols (100 each) and 4 small (10 each). Three clues link Cyan, Plum, and Lime to one another. We find Lime's amount.

Givens

Teal (from figure) = 6 large + 4 small symbols; 1 large = 100, 1 small = 10 tickets.
Cyan = Teal - 40.
Plum = $\frac{2}{3}$ of Cyan.
Lime = Plum + 130.

Unknowns

The amount for the Lime item.

Constraints

Amounts are whole numbers.
For Plum to be a whole number, Cyan must be a multiple of 3.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems

Each clue depends on the previous one, so we solve in order (Teal -> Cyan -> Plum -> Lime), each a small subproblem, chaining toward Lime.

3 · Execute4 carry out the plan

1Read Teal from the pictograph

#7 Identify Subproblems 3.MD.B.3

Teal shows 6 large symbols and 4 small ones. Each large is 100 and each small is 10: 6 x 100 + 4 x 10 = 640.

6 \times 100 + 4 \times 10 = 640

Turning the scaled symbols into a count is the core skill of a scaled pictograph.

2Find Cyan

#11 Work Backwards 3.NBT.A.2

Cyan has 40 fewer than Teal: 640 - 40 = 600. (This is a multiple of 3, so the next fraction step gives a whole number.)

640 - 40 = 600

Subtracting within the thousands is Grade 3 place-value work.

3Find Plum ( $\frac{2}{3}$ of Cyan)

#7 Identify Subproblems 3.NF.A.1

Plum is

\frac{2}{3}

of Cyan. Splitting 600 into 3 equal parts gives 600 / 3 = 200, and taking 2 of them gives Plum = 2 x 200 = 400.

\tfrac{2}{3}\times 600 = 2 \times 200 = 400

Understanding

\frac{2}{3}

as 2 of the 3 equal parts of a whole is the Grade 3 meaning of a fraction.

4Find Lime

#11 Work Backwards 3.NBT.A.2

Lime has 130 more than Plum: 400 + 130 = 530.

400 + 130 = 530

Adding 130 to 400 is a clean within-1000 addition.

Answer: 530 tickets

4 · Reviewdoes it hold up?

Lime (530) follows from the chain: the $\frac{2}{3}$ clue forces Cyan to be a multiple of 3; 600 = 3 x 200, so Plum = 2 x 200 = 400 and Lime = 400 + 130 = 530, all whole numbers.

Another way: Work the fraction as a subproblem first: one part of 600 is 200, so $\frac{2}{3}$ is 2 x 200 = 400, then add 130 to reach 530 - same answer.

Standardsmin grade 3

3.MD.B.3 Draw and interpret scaled picture graphs and bar graphs — Reading Teal's value from the large and small symbols.
3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Interpreting $\frac{2}{3}$ of Cyan as 2 of its 3 equal parts.
3.NBT.A.2 Fluently add and subtract within 1000 — Subtracting 40 and adding 130 along the chain of clues.

💡Takeaway. Solve the clues one after another - read the graph, then 2 of the 3 equal parts, then add 130!

Variant 8 hard

The amount for each item was surveyed and shown in a pictograph. Find the amount for the Reef item.

Amount by Item (pictograph)

Item	Amount
Bay
Cove	(4 large pictures, 8 small pictures)
Reef
Isle

In the pictograph, each large picture stands for $100\ \text{boats}$ and each small picture stands for $10\ \text{boats}$ .

Conditions

The Bay item has $60$ fewer than the Cove item.
The Isle item has $\dfrac{2}{3}$ as many as the Bay item.
The Reef item has $150$ more than the Isle item.

Your answer

Show solution

1 · Understandwhat's really being asked

A pictograph gives Cove as 4 large picture symbols (100 each) and 8 small (10 each). Three clues link Bay, Isle, and Reef to one another. We find Reef's amount.

Givens

Cove (from figure) = 4 large + 8 small symbols; 1 large = 100, 1 small = 10 boats.
Bay = Cove - 60.
Isle = $\frac{2}{3}$ of Bay.
Reef = Isle + 150.

Unknowns

The amount for the Reef item.

Constraints

Amounts are whole numbers.
For Isle to be a whole number, Bay must be a multiple of 3.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems

Each clue depends on the previous one, so we solve in order (Cove -> Bay -> Isle -> Reef), each a small subproblem, chaining toward Reef.

3 · Execute4 carry out the plan

1Read Cove from the pictograph

#7 Identify Subproblems 3.MD.B.3

Cove shows 4 large symbols and 8 small ones. Each large is 100 and each small is 10: 4 x 100 + 8 x 10 = 480.

4 \times 100 + 8 \times 10 = 480

Turning the scaled symbols into a count is the core skill of a scaled pictograph.

2Find Bay

#11 Work Backwards 3.NBT.A.2

Bay has 60 fewer than Cove: 480 - 60 = 420. (This is a multiple of 3, so the next fraction step gives a whole number.)

480 - 60 = 420

Subtracting within the thousands is Grade 3 place-value work.

3Find Isle ( $\frac{2}{3}$ of Bay)

#7 Identify Subproblems 3.NF.A.1

Isle is

\frac{2}{3}

of Bay. Splitting 420 into 3 equal parts gives

\frac{420}{3}

= 140, and taking 2 of them gives Isle = 2 x 140 = 280.

\tfrac{2}{3}\times 420 = 2 \times 140 = 280

Understanding

\frac{2}{3}

as 2 of the 3 equal parts of a whole is the Grade 3 meaning of a fraction.

4Find Reef

#11 Work Backwards 3.NBT.A.2

Reef has 150 more than Isle: 280 + 150 = 430.

280 + 150 = 430

Adding 150 to 280 is a clean within-1000 addition.

Answer: 430 boats

4 · Reviewdoes it hold up?

Reef (430) follows from the chain: the $\frac{2}{3}$ clue forces Bay to be a multiple of 3; 420 = 3 x 140, so Isle = 2 x 140 = 280 and Reef = 280 + 150 = 430, all whole numbers.

Another way: Work the fraction as a subproblem first: one part of 420 is 140, so $\frac{2}{3}$ is 2 x 140 = 280, then add 150 to reach 430 - same answer.

Standardsmin grade 3

3.MD.B.3 Draw and interpret scaled picture graphs and bar graphs — Reading Cove's value from the large and small symbols.
3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Interpreting $\frac{2}{3}$ of Bay as 2 of its 3 equal parts.
3.NBT.A.2 Fluently add and subtract within 1000 — Subtracting 60 and adding 150 along the chain of clues.

💡Takeaway. Solve the clues one after another - read the graph, then 2 of the 3 equal parts, then add 150!

Where next?

← Back to the problem: Solve missing data from ratio or sum clues