# 8.3: Circle Sums 6

Difficulty Level: At Grade Created by: CK-12

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 four letters? In this section, we will practice solving for unknowns in circles and arrows diagrams.

### Circle Sums 6

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 all seven of the arrows.

#### Finding Unknown Values

1. The arrows point to the sum of each row or column. Figure out the value of each letter.

We can use problem solving steps to help.

\begin{align*}& \mathbf{Describe:} && \text{There are rows and columns with numbers and letters.}\\ &&& \text{Arrows point to sums.}\\ &&& \text{There are four different letters.}\\ \\ & \mathbf{My \ Job:} && \text{Figure out the value of each letter,} \ A, \ B, \ C, \ \text{and} \ D.\\ \\ & \mathbf{Plan:} && \text{All of Column 1 is contained in Row 1.}\\ &&& \text{Replace} \ B, \ B, \ \text{and} \ D \ \text{in Row 1 with its value from Column 1. Find the value of} \ D.\\ &&& \text{Replace each} \ D \ \text{with its value to find the value of} \ B \ \text{in Column 1.}\\ &&& \text{Replace} \ B \ \text{with its value in Column 4 to find the value of} \ A.\\ &&& \text{Use Column 2 to find the value of} \ C. \\ \\ & \mathbf{Solve:} && \text{Column} \ 1: \ B+B+D=17.\\ &&& \text{Replace} \ B+B+D \ \text{with 17 in Row 1 to get} \ 17+D=24.\\ &&& D=7. \ \text{Replace each} \ D \ \text{with 7 everywhere it appears in the grid.}\\ &&& \text{Column} \ 1: \ B+B+7=17 \ \text{so} \ B=5.\ \text{Replace all B's with 5.}\\ &&& \text{Column} \ 4: \ 7+A+5=16 \ \text{so} \ A =4. \ \text{Replace all A's with 4.}\\ &&& \text{Column} \ 2: \ 5+C+4=17. \ \text{so} \ C = 8. \\ &&& \text{So} \ A =4, \ B =5, \ C = 8, \ \text{and} \ D =7.\\ \\ & \mathbf{Check:} && \text{Check rows and columns}\\ &&& \text{Row} \ 1: 5+5+7+7=24 \qquad \ \text{Column} \ 1: 5+5+7=17\\ &&& \text{Row} \ 2: 5+8+4+4=21 \qquad \ \text{Column} \ 2: 5+8+4=17\\ &&& \text{Row} \ 3: 7+4+4+5=20 \qquad \ \text{Column} \ 3: 7+4+4=15\\ &&& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \text{Column} \ 4: 7+4+5=16\end{align*}

2. The arrows point to the sum of each row or column. Figure out the value of each letter.

We can use problem solving steps to help.

\begin{align*}& \mathbf{Describe:} && \text{There are rows and columns with numbers and letters.}\\ &&& \text{Arrows point to sums.}\\ &&& \text{There are four different letters.}\\ \\ & \mathbf{My \ Job:} && \text{Figure out the value of each letter,} \ A, \ B, \ C, \ \text{and} \ D.\\ \\ & \mathbf{Plan:} && \text{All of Column 1 is contained in Row 1.}\\ &&& \text{Replace} \ C, \ C, \ \text{and} \ D \ \text{in Row 1 with its value from Column 1. Find the value of} \ D.\\ &&& \text{Replace each} \ D \ \text{with its value to find the value of} \ C \ \text{in Column 1.}\\ &&& \text{Replace} \ C \ \text{with its value in Column 2 to find the value of} \ A.\\ &&& \text{Use Column 3 to find the value of} \ B. \\ \\ & \mathbf{Solve:} && \text{Column} \ 1: \ C+C+D=17.\\ &&& \text{Replace} \ C+C+D \ \text{with 17 in Row 1 to get} \ 17+D=26.\\ &&& D=9. \ \text{Replace each} \ D \ \text{with 9 everywhere it appears in the grid.}\\ &&& \text{Column} \ 1: \ C+C+9=17 \ \text{so} \ C=4.\ \text{Replace all C's with 4.}\\ &&& \text{Column} \ 2: \ 4+9+A=19 \ \text{so} \ A =6. \ \text{Replace all A's with 6.}\\ &&& \text{Column} \ 3: \ 9+6+B=22. \ \text{so} \ B = 7. \\ &&& \text{So} \ A =6, \ B =7, \ C = 4, \ \text{and} \ D =9.\\ \\ & \mathbf{Check:} && \text{Check rows and columns}\\ &&& \text{Row} \ 1: 4+4+9+9=26 \qquad \ \text{Column} \ 1: 4+4+9=17\\ &&& \text{Row} \ 2: 4+9+6+6=25 \qquad \ \text{Column} \ 2: 4+9+6=19\\ &&& \text{Row} \ 3: 9+6+7+4=26 \qquad \ \text{Column} \ 3: 9+6+7=22\\ &&& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \text{Column} \ 4: 9+6+4=19\end{align*}

3. The arrows point to the sum of each row or column. Figure out the value of each letter.

We can use problem solving steps to help.

\begin{align*}& \mathbf{Describe:} && \text{There are rows and columns with numbers and letters.}\\ &&& \text{Arrows point to sums.}\\ &&& \text{There are four different letters.}\\ \\ & \mathbf{My \ Job:} && \text{Figure out the value of each letter,} \ A, \ B, \ C, \ \text{and} \ D.\\ \\ & \mathbf{Plan:} && \text{All of Column 3 is contained in Row 1.}\\ &&& \text{Replace} \ A, \ C, \ \text{and} \ C \ \text{in Row 1 with its value from Column 3. Find the value of} \ D.\\ &&& \text{Replace each} \ D \ \text{with its value to find the value of} \ A \ \text{in Column 2.}\\ &&& \text{Replace} \ A \ \text{with its value in Column 3 to find the value of} \ C.\\ &&& \text{Use Column 1 to find the value of} \ B. \\ \\ & \mathbf{Solve:} && \text{Column} \ 3: \ C+A+C =13.\\ &&& \text{Replace} \ A+C+C \ \text{with 13 in Row 1 to get} \ D+13 =19.\\ &&& D=6. \ \text{Replace each} \ D \ \text{with 6 everywhere it appears in the grid.}\\ &&& \text{Column} \ 2: \ A+6+6=19 \ \text{so} \ A=7.\ \text{Replace all A's with 7.}\\ &&& \text{Column} \ 3: \ C+7+C=13 \ \text{so} \ C =3. \ \text{Replace all C's with 3.}\\ &&& \text{Column} \ 1: \ 6+B+7=21. \ \text{so} \ B = 8. \\ &&& \text{So} \ A =7, \ B =8, \ C = 3, \ \text{and} \ D =6.\\ \\ & \mathbf{Check:} && \text{Check rows and columns}\\ &&& \text{Row} \ 1: 6+7+3+3=19 \qquad \ \text{Column} \ 1: 6+8+7=21\\ &&& \text{Row} \ 2: 8+6+7+7=28 \qquad \ \text{Column} \ 2: 7+6+6=19\\ &&& \text{Row} \ 3: 7+6+3+7=23 \qquad \ \text{Column} \ 3: 3+7+3=13\\ &&& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \text{Column} \ 4: 3+7+7=17\end{align*}

#### Earlier Problem Revisited

We can use problem solving steps to help.

\begin{align*}& \mathbf{Describe:} && \text{There are rows and columns with numbers and letters.}\\ &&& \text{Arrows point to sums.}\\ &&& \text{There are four different letters.}\\ \\ & \mathbf{My \ Job:} && \text{Figure out the value of each letter,} \ A, \ B, \ C, \ \text{and} \ D.\\ \\ & \mathbf{Plan:} && \text{All of Column 1 is contained in Row 1.}\\ &&& \text{Replace} \ A, \ B, \ \text{and} \ C \ \text{in Row 1 with its value from Column 1. Find the value of} \ C.\\ &&& \text{Replace each} \ C \ \text{with its value to find the value of} \ D \ \text{in Column 4, and} \ A \ \text{in}\\ &&& \text{Column 3. Replace} \ A \ \text{with its value in Column 2 to find the value of} \ B.\\ \\ & \mathbf{Solve:} && \text{Column} \ 1: \ A + B + C =14.\\ &&& \text{Replace} \ A + B + C \ \text{with 14 in Row 1 to get} \ 14 + C =19.\\ &&& C=5. \ \text{Replace each} \ C \ \text{with 5 everywhere it appears in the grid.}\\ &&& \text{Column} \ 4: \ 5 + 5 + D =1 \ \text{and} \ D =4.\ \text{Replace all D's with 4.}\\ &&& \text{Column} \ 3: \ 5 + 5 + A =13 \ \text{and} \ A =3. \ \text{Replace all A's with 3.}\\ &&& \text{Column} \ 2: \ B + B + 3 = 15. \ \text{Then} \ 2B = 12, \ \text{and} \ B =6.\\ &&& \text{So} \ A =3, \ B =6, \ C = 5, \ \text{and} \ D =4.\\ \\ & \mathbf{Check:} && \text{Check rows and columns}\\ &&& \text{Row} \ 1: 3 + 6 + 5 + 5 = 19 \qquad \ \text{Column} \ 1: 3 + 6 + 5 = 14\\ &&& \text{Row} \ 2: 6 + 6 + 5 + 5 = 22 \qquad \ \text{Column} \ 2: 6 + 6 + 3 = 15\\ &&& \text{Row} \ 3: 5 + 3 + 3 + 4 = 15 \qquad \ \text{Column} \ 3: 5 + 5 + 3 = 13\\ &&& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \text{Column} \ 4: 5 + 5 + 4 = 14\end{align*}

### 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.

### Examples

#### Example 1

The arrows point to the sum of each row or column. Figure out the value of each letter.

\begin{align*}A = 5, \ B = 9, \ C = 7, \ D = 8\end{align*}

#### Example 2

The arrows point to the sum of each row or column. Figure out the value of each letter.

\begin{align*}A = 9, \ B = 6, \ C = 7, \ D = 3\end{align*}

#### Example 3

The arrows point to the sum of each row or column. Figure out the value of each letter.

\begin{align*}A = 8, \ B = 7, \ C = 5, \ D = 9\end{align*}

### Review

For each problem, the arrows point to the sum of each row or column. Figure out the value of each letter.

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Jan 18, 2013