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# 6.2: Circles and Arrows 4

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 three 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 three letters in the diagram.
• Then, make a plan for how you will solve. There is usually more than one way to solve the problem. You will want to figure out one letter first and then the next letter.
• Next, solve the problem.
• Finally, check to make sure that the values you found work for all six of the arrows.

#### Example A

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

Solution:

We can use the 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 three different letters}.\!\\ & \mathbf{Job} && \text{Figure out the value of each letter,} \ D, E, \ \text{and} \ F.\!\\ & \mathbf{Plan} && \text{In the second row, there are} \ 2 \ \text{Ds and a} \ 2.\!\\ &&& \text{Start with that row.}\!\\ & \mathbf{Solve} && \text{Row} \ 2: \ D+D+2=10, \ \text{so} \ B + B = 8 \ \text{and} \ D = 4.\!\\ &&& \text{Column} \ 1 \ \text{has} \ 2 \ \text{Es and a} \ D. \ \text{Replace} \ D \ \text{with} \ 4 \ \text{and solve for} \ E, E+D+E=6, \ \text{so} \!\\ &&& E+4+E=6, \ E+E=2, \ \text{and} \ E = 1.\!\\ &&& \text{Row} \ 3: \ \text{replace} \ E \ \text{and} \ D \ \text{with their values, and find} \ F. \ 1+4+F = 15, \ \text{so} \!\\ &&& 5+F=15, \ \text{and} \ F = 10.\!\\ & \mathbf{Check} && \text{Rows top to bottom}: \ \ 1+1+10=12 \ \text{Columns left to right}: \ 1+4+1=6\!\\ &&& \qquad \qquad \qquad \qquad \qquad \ 4+4+2=10 \qquad \qquad \qquad \qquad \qquad \ \ 1+4+4=9\!\\ &&& \qquad \qquad \qquad \qquad \qquad \ 1+4+10=15 \qquad \qquad \qquad \qquad \qquad \ \ 10+2+10=22\end{align*}

#### Example B

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

Solution:

We can use the 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 three different letters}.\!\\ & \mathbf{Job} && \text{Figure out the value of each letter,} \ J, H, \ \text{and} \ G.\!\\ & \mathbf{Plan} && \text{In the third column, there are} \ 2 \ \text{Gs and a} \ 5.\!\\ &&& \text{Start with that column.}\!\\ & \mathbf{Solve} && \text{Column} \ 3: \ G+5+G=19, \ \text{so} \ G+G=14 \ \text{and} \ G=7.\!\\ &&& \text{Row} \ 3 \ \text{has} \ 2 \ \text{Gs and a} \ J. \ \text{Replace} \ G \ \text{with} \ 7 \ \text{and solve for} \ J, 7+J+7=20, \ \text{so} \!\\ &&& J+14=20 \ \text{and} \ J=6.\!\\ &&& \text{Column} \ 2: \ \text{replace} \ J \ \text{with its value, and find} \ H. \ H+H+6=12, \ \text{so} \!\\ &&& H+H=6, \ \text{and} \ H=3.\!\\ & \mathbf{Check} && \text{Rows top to bottom}: \ \ 6+3+7=16 \ \text{Columns left to right}: \ 6+6+7=19\!\\ &&& \qquad \qquad \qquad \qquad \qquad \ 6+3+5=14 \qquad \qquad \qquad \qquad \qquad \ \ 3+3+6=12\!\\ &&& \qquad \qquad \qquad \qquad \qquad \ 7+6+7=20 \qquad \qquad \qquad \qquad \qquad \ \ 7+5+7=19\end{align*}

#### Example C

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

Solution:

We can use the 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 three different letters}.\!\\ & \mathbf{Job} && \text{Figure out the value of each letter,} \ K, M, \ \text{and} \ N.\!\\ & \mathbf{Plan} && \text{In the first column, there are} \ 2 \ \text{Ks and a} \ 4.\!\\ &&& \text{Start with that column.}\!\\ & \mathbf{Solve} && \text{Column} \ 1: \ 4+K+K=20, \ \text{so} \ K+K=16 \ \text{and} \ K=8.\!\\ &&& \text{Row} \ 2 \ \text{has} \ 2 \text{Ks and an} \ M. \ \text{Replace} \ K \ \text{with} \ 8 \ \text{and solve for} \ M, 8+M+8=20, \ \text{so} \!\\ &&& 16+M=20 \ \text{and} \ M=4.\!\\ &&& \text{Column} \ 3: \ \text{replace} \ K \ \text{and} \ M \ \text{with their values, and find} \ N. \ N+8+4=15, \ \text{so} \!\\ &&& N+12=15, \ \text{and} \ N=3.\!\\ & \mathbf{Check} && \text{Rows top to bottom}: \ \ 4+4+3=11 \ \text{Columns left to right}: \ 4+8+8=20\!\\ &&& \qquad \qquad \qquad \qquad \qquad \ 8+4+8=20 \qquad \qquad \qquad \qquad \qquad \ \ 4+4+3=11\!\\ &&& \qquad \qquad \qquad \qquad \qquad \ 8+3+4=15 \qquad \qquad \qquad \qquad \qquad \ \ 3+8+4=15\end{align*}

#### Concept Problem Revisited

We can use the 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 three different letters}.\!\\ & \mathbf{Job} && \text{Figure out the value of each letter,} \ A, B, \ \text{and} \ C.\!\\ & \mathbf{Plan} && \text{In the first column, there are} \ 2 \ \text{Bs and a} \ 6.\!\\ &&& \text{Start with that column.}\!\\ & \mathbf{Solve} && \text{Column} \ 1: \ 6 + B + B = 16, \ \text{so} \ B + B = 10 \ \text{and} \ B = 5.\!\\ &&& \text{Row} \ 1 \ \text{has} \ 6, B \ \text{and} \ A. \ \text{Replace} \ B \ \text{with} \ 5 \ \text{and solve for} \ A, 6 + A + 5 = 13, \ \text{so} \!\\ &&& A + 11 =13 \ \text{and} \ A = 2.\!\\ &&& \text{Column} \ 2: \ \text{replace} \ A \ \text{and} \ B \ \text{with their values, and find} \ C. \ 2 + 5 + C = 15, \ \text{so} \!\\ &&& 7 + C=15, \ \text{and} \ C = 8.\!\\ & \mathbf{Check} && \text{Rows top to bottom}: \ \ 6 + 2 + 5 = 13 \ \text{Columns left to right}: \ 6 + 5 + 5 = 16\!\\ &&& \qquad \qquad \qquad \qquad \qquad \ 5 + 5 + 8 = 18 \qquad \qquad \qquad \qquad \qquad \ \ 2 + 5 + 8 = 15\!\\ &&& \qquad \qquad \qquad \qquad \qquad \ 5 + 8 + 2 = 15 \qquad \qquad \qquad \qquad \qquad \ \ 5 + 8 + 2 = 15\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.

### Guided Practice

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

1.

2.

3.

Answers:

1. \begin{align*}P = 6, \ R = 2, \ S = 8\end{align*}

2. \begin{align*}Q = 5, \ T = 7, \ U =4\end{align*}

3. \begin{align*}V = 3, \ W = 9, \ Z = 5\end{align*}

### Practice

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

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Difficulty Level:
At Grade
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Date Created:
Jan 18, 2013
Last Modified:
Mar 23, 2016

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