A Story of Numbers | FIO

Question 15

Add the following numerals that are in the base-5 system that we created:

Remember that in this system, 5 times a landmark number gives the next one!

Solution

Understand the Question

In this base-5 system, each landmark symbol represents a power of $5$ $5$ :
- Triangle $(\triangle) = 5^0 = 1$
- Square $(\square) = 5^1 = 5$
- Hexagon $(\hexagon) = 5^2 = 25$
- Circle $(\bigcirc) = 5^3 = 125$
To find the sum:
1. Convert both shape numerals to decimal (base-10) values.
2. Add the base-10 values together.
3. Convert the total sum back into base-5 symbols by finding how many of each landmark number fit into the sum (from largest to smallest).

Step 1 · Assign Values to Symbols

In this system, 5 times a landmark number gives the next one. The symbols represent powers of 5: Diagram 1

$\text{Value of Triangle }(\triangle) = 5^0 = 1$

$\text{Value of Square }(\square) = 5^1 = 5$

$\text{Value of Hexagon }(\hexagon) = 5^2 = 25$

$\text{Value of Circle }(\bigcirc) = 5^3 = 125$

Step 2 · Convert First Number to Base-10

The first number is $\bigcirc \hexagon \hexagon \square \triangle \triangle$ .

Counting the symbols: $\text{Number of Circles} = 1$ $\text{Number of Hexagons} = 2$ $\text{Number of Squares} = 1$ $\text{Number of Triangles} = 2$

Calculate the total value:

\begin{aligned} \text{First Number} &= (1 \times 125) + (2 \times 25) + (1 \times 5) + (2 \times 1) \\[0.6em] &= 125 + 50 + 5 + 2 \\[0.6em] &= 182 \end{aligned}

Step 3 · Convert Second Number to Base-10

The second number is $\bigcirc \bigcirc \bigcirc \hexagon \square \square \triangle \triangle$ .

Counting the symbols: $\text{Number of Circles} = 3$ $\text{Number of Hexagons} = 1$ $\text{Number of Squares} = 2$ $\text{Number of Triangles} = 2$

Calculate the total value:

\begin{aligned} \text{Second Number} &= (3 \times 125) + (1 \times 25) + (2 \times 5) + (2 \times 1) \\[0.6em] &= 375 + 25 + 10 + 2 \\[0.6em] &= 412 \end{aligned}

Step 4 · Add the Two Numbers

Add the decimal values of both numbers:

\begin{aligned} \text{Sum} &= \text{First Number} + \text{Second Number} \\[0.6em] &= 182 + 412 \\[0.6em] &= 594 \end{aligned}

Step 5 · Convert Sum Back to Base-5 Symbols

Represent $594$ using the base-5 symbols ( $\bigcirc = 125$ , $\hexagon = 25$ , $\square = 5$ , $\triangle = 1$ ) starting from the largest:

Circles ( $125$ ):

\begin{aligned} \text{Number of Circles} &= 594 \div 125 \\[0.6em] &= 4 \text{ with a remainder of } 94 \end{aligned}

Use 4 Circles ( $\bigcirc \bigcirc \bigcirc \bigcirc$ ).

Hexagons ( $25$ ):

\begin{aligned} \text{Number of Hexagons} &= 94 \div 25 \\[0.6em] &= 3 \text{ with a remainder of } 19 \end{aligned}

Use 3 Hexagons ( $\hexagon \hexagon \hexagon$ ).

Squares ( $5$ ):

\begin{aligned} \text{Number of Squares} &= 19 \div 5 \\[0.6em] &= 3 \text{ with a remainder of } 4 \end{aligned}

Use 3 Squares ( $\square \square \square$ ).

Triangles ( $1$ ):

\begin{aligned} \text{Number of Triangles} &= 4 \div 1 \\[0.6em] &= 4 \text{ with a remainder of } 0 \end{aligned}

Use 4 Triangles ( $\triangle \triangle \triangle \triangle$ ).

Combining the symbols gives: $\bigcirc \bigcirc \bigcirc \bigcirc \hexagon \hexagon \hexagon \square \square \square \triangle \triangle \triangle \triangle$

Answer

$\bigcirc \bigcirc \bigcirc \bigcirc \hexagon \hexagon \hexagon \square \square \square \triangle \triangle \triangle \triangle$ ( $594$ in base-10)

Common Mistakes

Base Confusion: Treating place values as powers of $10$ instead of powers of $5$ ( $1, 5, 25, 125$ ).
Unreduced Symbols: Leaving $5$ or more of any symbol in the final answer instead of regrouping them into the next higher landmark symbol (e.g., $5$ triangles must be regrouped into $1$ square).
Remainder Tracking: Making arithmetic errors when carrying remainders to smaller symbol values during division.

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