# 8.4: Circle Sums

**At Grade**Created by: CK-12

**Circle Sums – Solve for Values of Unknowns**

**Teacher Notes**

Circle Sums show grids in which circles contain letters that represent unknowns. Arrows point to sums of rows and columns. For example, in Row 1, \begin{align*}A + B + C + C = 19\end{align*}, and in Column 1, \begin{align*}A + B + C = 14\end{align*}. Students use the grid relationships to figure out the values of the letters. Each grid contains four different letters.

**Solutions:**

- \begin{align*}A = 4, \ B = 5, \ C = 8, \ D = 7\end{align*}
- \begin{align*}A = 6, \ B= 7, \ C = 4, \ D = 9\end{align*}
- \begin{align*}A = 7, \ B= 8, \ C = 3, \ D = 6\end{align*}
- \begin{align*}A = 5, \ B = 9, \ C = 7, \ D = 8\end{align*}

**Circle Sums – Solve for Values of Unknowns**

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

\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{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 or 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} \ 1: 3 + 6 + 5 = 14\\ &&& \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*}

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

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

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

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

*Extra for Experts:***Circle Sums – Solve for Values of Unknowns**

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

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

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

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

**Solutions:**

- \begin{align*}A = 9, \ B = 6, \ C = 7, \ D = 3\end{align*}
- \begin{align*}A = 8, \ B = 7, \ C = 5, \ D = 9\end{align*}
- \begin{align*}A = 6, \ B = 5, \ C = 8, \ D = 7\end{align*}
- \begin{align*}A = 7, \ B = 8, \ C = 9, \ D = 5\end{align*}

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