FIO-15

Chapter: A STORY OF NUMBERS
Class: 8 (Class 8)
Category: figure_it_out

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Question

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!

Question diagram(s):

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We will first figure out the decimal value for each shape. Then, we will convert the given numbers from shapes to base-10 numbers. After adding them, we will convert the total sum back into the shape system.

Step 1 — Assigning Values to Symbols

The problem tells us that in this system, 5 times a landmark number gives the next one. This means our symbols represent powers of 5. Let us assign the smallest value, $5^0$, to the triangle.

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

The next value is $1 \times 5 = 5$. We assign this to the square.

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

The next value is $5 \times 5 = 25$. We assign this to the hexagon.

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

The next value is $25 \times 5 = 125$. We assign this to the circle.

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

Step 2 — Converting the First Number to Base-10

The first number is shown as $\bigcirc \hexagon \hexagon \square \triangle \triangle$. We count how many of each symbol are present in this number.

$\text{Number of Circles } = 1$

$\text{Number of Hexagons } = 2$

$\text{Number of Squares } = 1$

$\text{Number of Triangles } = 2$

Now, we calculate the total value by adding up the values of all these symbols.

$\text{First Number} = (1 \times 125) + (2 \times 25) + (1 \times 5) + (2 \times 1)$

$= 125 + 50 + 5 + 2$

$\boxed{\text{First Number} = 182}$

Step 3 — Converting the Second Number to Base-10

The second number is shown as $\bigcirc \bigcirc \bigcirc \hexagon \square \square \triangle \triangle$. We count how many of each symbol are present in this number.

$\text{Number of Circles } = 3$

$\text{Number of Hexagons } = 1$

$\text{Number of Squares } = 2$

$\text{Number of Triangles } = 2$

Now, we calculate the total value by adding up the values of all these symbols.

$\text{Second Number} = (3 \times 125) + (1 \times 25) + (2 \times 5) + (2 \times 1)$

$= 375 + 25 + 10 + 2$

$\boxed{\text{Second Number} = 412}$

Step 4 — Adding the Two Numbers

We add the base-10 values of the first number and the second number.

$\text{Sum} = \text{First Number} + \text{Second Number}$

$= 182 + 412$

$\boxed{\text{Sum} = 594}$

Step 5 — Converting the Sum Back to Base-5 Symbols

We need to represent the sum, 594, using our base-5 symbols ($\bigcirc=125$, $\hexagon=25$, $\square=5$, $\triangle=1$). We start with the largest value and see how many times it fits into the sum.

First, we find how many Circles (125) are in 594.

$\text{Number of Circles} = 594 \div 125$

$= 4 \text{ with a remainder of } 94$

So, we use 4 Circles ($\bigcirc \bigcirc \bigcirc \bigcirc$). The remaining value is 94.

Next, we find how many Hexagons (25) are in the remaining value of 94.

$\text{Number of Hexagons} = 94 \div 25$

$= 3 \text{ with a remainder of } 19$

So, we use 3 Hexagons ($\hexagon \hexagon \hexagon$). The remaining value is 19.

Next, we find how many Squares (5) are in the remaining value of 19.

$\text{Number of Squares} = 19 \div 5$

$= 3 \text{ with a remainder of } 4$

So, we use 3 Squares ($\square \square \square$). The remaining value is 4.

Finally, we find how many Triangles (1) are in the remaining value of 4.

$\text{Number of Triangles} = 4 \div 1$

$= 4 \text{ with a remainder of } 0$

So, we use 4 Triangles ($\triangle \triangle \triangle \triangle$). There is no remainder left.

Combining all the symbols, the sum 594 is represented as: $\bigcirc \bigcirc \bigcirc \bigcirc \hexagon \hexagon \hexagon \square \square \square \triangle \triangle \triangle \triangle$

Answer

(i) The first number in base-10 is 182. (ii) The second number in base-10 is 412. (iii) The sum in base-10 is 594, which is represented as $\bigcirc \bigcirc \bigcirc \bigcirc \hexagon \hexagon \hexagon \square \square \square \triangle \triangle \triangle \triangle$ in base-5 symbols.