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Look at the picture below. The arrows point to the sum of each row or column. Can you figure out the value of each of the five letters? In this concept, we will practice solving for unknowns in circles and arrows diagrams.

Guidance

In order to solve the problem above, use the problem solving steps.

  • Start by describing what you see in the diagram.
  • Next, figure out what your job is in this problem. In all of these problems your job will be to figure out the value of the four letters in the diagram.
  • Then, make a plan for how you will solve. There is usually more than one way to solve the problem. In some of these problems you might need to find the sum of three letters first in order to find the value of one letter.
  • Next, solve the problem.
  • Finally, check to make sure that the values you found work for the rows or the columns.

Example A

Letters stand for numbers. Same letters are same numbers. Numbers in squares are row and column sums. Arrows point to the sums.

What are the values? \mathbf{M} = \underline{\;\;\;\;\;\;\;\;} \ \mathbf{P} = \underline{\;\;\;\;\;\;\;\;} \ \mathbf{Q} = \underline{\;\;\;\;\;\;\;\;} \ \mathbf{T} = \underline{\;\;\;\;\;\;\;\;} \ \mathbf{W} = \underline{\;\;\;\;\;\;\;\;}

Solution:

We can use problem solving steps to help us to find the values of the letters.

&\mathbf{Describe} && \text{There are three rows and four columns.}\\&&& \text{Circles have letters or numbers.}\\&&& \text{Sums of rows and columns are in the squares.}\\&&& \text{Arrows point to sums.}\\\\& \mathbf{My \ Job} && \text{Figure out values of letters,} \ M, P, Q, T, \ \text{and} \ W.\\\\&\mathbf{Plan} && \text{Look for a column that contains all letters shown in a row. Replace the group of}\\&&& \text{same letters with its value and continue.}\\\\&\mathbf{Solve} && \text{All of Column} \ 1 \ \text{is contained in Row} \ 2. \ \text{Replace} \ TWQ \ \text{in Row} \ 2 \ \text{with} \ 28. \ \text{Then} \ 28+M=41,\\&&& \text{and} \ M = 13. \ \text{Replace all} \ M' \text{s in the grid with} \ 13.\\&&& \text{Column} \ 2: P+13+13=36. \ \text{So}, \ P=10. \ \text{Replace all} \ P' \text{s with} \ 10.\\&&& \text{Row} \ 1: W+10+4+W=18. \ \text{So}, \ 2W=4 \ \text{and} \ W=2 \ \text{Replace} \ W' \text{s with} \ 2.\\&&& \text{Row} \ 3: Q+13+Q+10=63. \ \text{So}, \ 2Q=40 \ \text{and} \ Q = 20. \ \text{Replace} \ Q'\text{s with} \ 20.\\&&& \text{Column} \ 1: 2+T+20=28. \ \text{So}, \ T+22=28, \ \text{and} \ T=6. \ \text{Replace} \ T \ \text{with} \ 6.\\\\&\mathbf{Check} && \text{Check rows or columns}\\&&& \text{Row} \ 1: 2+10+4+2=18\\&&& \text{Row} \ 2: 6+13+2+20=41\\&&& \text{Row} \ 3: 20+13+20+10=63

Example B

Letters stand for numbers. Same letters are same numbers. Numbers in squares are row and column sums. Arrows point to the sums.

What are the values? \mathbf{C} = \underline{\;\;\;\;\;\;\;\;} \ \mathbf{D} = \underline{\;\;\;\;\;\;\;\;} \ \mathbf{E} = \underline{\;\;\;\;\;\;\;\;} \ \mathbf{F} = \underline{\;\;\;\;\;\;\;\;} \ \mathbf{G} = \underline{\;\;\;\;\;\;\;\;}

Solution:

We can use problem solving steps to help us to find the values of the letters.

&\mathbf{Describe} && \text{There are three rows and four columns.}\\&&& \text{Circles have letters or numbers.}\\&&& \text{Sums of rows and columns are in the squares.}\\&&& \text{Arrows point to sums.}\\\\& \mathbf{My \ Job} && \text{Figure out values of letters,} \ C, D, E, F, \ \text{and} \ G.\\\\&\mathbf{Plan} && \text{Look for a column that contains all letters shown in a row. Replace the group of}\\&&& \text{same letters with its value and continue.}\\\\&\mathbf{Solve} && \text{All of Column} \ 3 \ \text{is contained in Row} \ 3. \ \text{Replace} \ GDE \ \text{in Row} \ 3 \ \text{with} \ 17. \ \text{Then} \ 17+F=22,\\&&& \text{and} \ F = 5. \ \text{Replace all} \ F' \text{s in the grid with} \ 5.\\&&& \text{Column} \ 4: D+D+5=13. \ \text{So}, \ 2D = 8 \ \text{and} \ D = 4. \ \text{Replace all} \ D' \text{s with} \ 4.\\&&& \text{Row} \ 1: C+C+4+4=12. \ \text{So}, \ 2C=4 \ \text{and} \ C=2 \ \text{Replace} \ C' \text{s with} \ 2.\\&&& \text{Row} \ 3: G+4+7+5=22. \ \text{So}, \ G+16=22 \ \text{and} \ G = 6. \ \text{Replace} \ G'\text{s with} \ 6.\\&&& \text{Column} \ 1: 2+E+6=15. \ \text{So}, \ E+8=15, \ \text{and} \ E=7. \ \text{Replace} \ E \ \text{with} \ 7.\\\\&\mathbf{Check} && \text{Check rows or columns}\\&&& \text{Row} \ 1: 2+2+4+4=12\\&&& \text{Row} \ 2: 7+8+6+4=25\\&&& \text{Row} \ 3: 6+4+7+5=22

Example C

Letters stand for numbers. Same letters are same numbers. Numbers in squares are row and column sums. Arrows point to the sums.

What are the values? \mathbf{A} = \underline{\;\;\;\;\;\;\;\;} \ \mathbf{B} = \underline{\;\;\;\;\;\;\;\;} \ \mathbf{X} = \underline{\;\;\;\;\;\;\;\;} \ \mathbf{Y} = \underline{\;\;\;\;\;\;\;\;} \ \mathbf{Z}= \underline{\;\;\;\;\;\;\;\;}

Solution:

We can use problem solving steps to help us to find the values of the letters.

&\mathbf{Describe} && \text{There are three rows and four columns.}\\&&& \text{Circles have letters or numbers.}\\&&& \text{Sums of rows and columns are in the squares.}\\&&& \text{Arrows point to sums.}\\\\& \mathbf{My \ Job} && \text{Figure out values of letters,} \ A, B, X, Y, \ \text{and} \ Z.\\\\&\mathbf{Plan} && \text{Look for a column that contains all letters shown in a row. Replace the group of}\\&&& \text{same letters with its value and continue.}\\\\&\mathbf{Solve} && \text{All of Column} \ 1 \ \text{is contained in Row} \ 1. \ \text{Replace} \ YZX \ \text{in Row} \ 1 \ \text{with} \ 27. \ \text{Then} \ 27+Y=39,\\&&& \text{and} \ Y = 12. \ \text{Replace all} \ Y' \text{s in the grid with} \ 12.\\&&& \text{Column} \ 4: 12+Z+Z=20. \ \text{So}, \ 2Z = 8 \ \text{and} \ Z = 4. \ \text{Replace all} \ Z' \text{s with} \ 4.\\&&& \text{Column} \ 1: 12+4+X = 27. \ \text{So}, \ 16+X = 27 \ \text{so} \ X = 11 \ \text{Replace} \ X' \text{s with} \ 11.\\&&& \text{Row} \ 3: 11+B+12+4=30. \ \text{So}, \ B + 27 = 30 \ \text{and} \ B = 3. \ \text{Replace} \ B'\text{s with} \ 3.\\&&& \text{Column} \ 2: 4+A+3=8. \ \text{So}, \ A + 7 = 8, \ \text{and} \ A = 1. \ \text{Replace} \ A \ \text{with} \ 1.\\\\&\mathbf{Check} && \text{Check rows or columns}\\&&& \text{Row} \ 1: 12+4+11+12=39\\&&& \text{Row} \ 2: 4+1+3+4=12\\&&& \text{Row} \ 3: 11+3+12+4=30

Concept Problem Revisited

We can use problem solving steps to help us to find the values of the letters.

&\mathbf{Describe} && \text{There are three rows and four columns.}\\&&& \text{Circles have letters or numbers.}\\&&& \text{Sums of rows and columns are in the squares.}\\&&& \text{Arrows point to sums.}\\\\& \mathbf{My \ Job} && \text{Figure out values of letters,} \ A, B, C, D, \ \text{and} \ E.\\\\&\mathbf{Plan} && \text{Look for a column that contains all letters shown in a row. Replace the group of}\\&&& \text{same letters with its value and continue.}\\\\&\mathbf{Solve} && \text{All of Column} \ 4 \ \text{is contained in Row} \ 3. \ \text{Replace} \ ABC \ \text{in Row} \ 3 \ \text{with} \ 18. \ \text{Then} \ 18 + C = 22,\\&&& \text{and} \ C = 4. \ \text{Replace all} \ C' \text{s in the grid with} \ 4.\\&&& \text{Column} \ 3: D + D + 4 = 22. \ \text{So}, \ 2D = 18 \ \text{and} \ D = 9. \ \text{Replace all} \ D' \text{s with} \ 9.\\&&& \text{Row} \ 1: 2 + A + 9 + A = 27. \ \text{So}, \ 2A + 11 = 27 \ \text{and} \ 2A = 16 \ \text{and} \ A = 8. \ \text{Replace} \ A' \text{s with} \ 8.\\&&& \text{Column} \ 4: 8 + B + 4 = 18. \ \text{So}, \ B + 12 = 18 \ \text{and} \ B = 6. \ \text{Replace} \ B'\text{s with} \ 6.\\&&& \text{Column} \ 1: 2 + E + 8 = 11. \ \text{So}, \ E + 10 = 11, \ \text{and} \ E = 1. \ \text{Replace} \ E \ \text{with} \ 1.\\\\&\mathbf{Check} && \text{Check rows or columns}\\&&& \text{Row} \ 1: 2 + 8 + 9 + 8 = 27\\&&& \text{Row} \ 2: 1 + 8 + 9 + 6 = 24\\&&& \text{Row} \ 3: 8 + 6 + 4 + 4 = 22

Vocabulary

In math, an unknown is a letter that stands for a number that we do not yet know the value of. In this concept, when you figured out the value of the letters in the circles and arrows diagrams you were solving for unknowns.

Guided Practice

In all problems, letters stand for numbers, same letters are same numbers, numbers in squares are row and column sums, arrows point to the sums. Find the values of the five letters.

1.

2.

3.

Answers:

1. E = 9, G = 7, L = 10, R = 5, S=4

2. D = 3, E = 8, F = 5, G = 15, H=20

3. S = 1, T = 20, U = 5, X = 12, W=9

Practice

In all problems, letters stand for numbers, same letters are same numbers, numbers in squares are row and column sums, arrows point to the sums. Find the values of the five letters.

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Difficulty Level:

At Grade

Grades:

Date Created:

Jan 18, 2013

Last Modified:

Nov 05, 2014
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