Question 11

We will express each number as a sum of powers of 5, using the given symbols.

Step 1 — Understanding the Symbols

Let us first list the value of each symbol from the provided table. The symbols represent powers of 5. $\triangle = 5^0 = 1$ $\square = 5^1 = 5$ $\hexagon = 5^2 = 25$ $\bigcirc = 5^3 = 125$ $\sim = 5^4 = 625$ $\uparrow = 5^5 = 3125$

$\boxed{\text{Each symbol represents a power of 5.}}$

Step 2 — Converting 15

We want to find how many of each power of 5 are in 15. The largest power of 5 less than or equal to 15 is $5^1 = 5$. We divide 15 by 5. $15 \div 5 = 3 \text{ with a remainder of } 0$ So, 15 is made up of three 5s. $15 = 3 \times 5$ We use three Square symbols for 15.

$\boxed{15 = \square \square \square}$

Step 3 — Converting 50

We want to find how many of each power of 5 are in 50. The largest power of 5 less than or equal to 50 is $5^2 = 25$. We divide 50 by 25. $50 \div 25 = 2 \text{ with a remainder of } 0$ So, 50 is made up of two 25s. $50 = 2 \times 25$ We use two Hexagon symbols for 50.

$\boxed{50 = \hexagon \hexagon}$

Step 4 — Converting 137

We want to find how many of each power of 5 are in 137. The largest power of 5 less than or equal to 137 is $5^3 = 125$. We divide 137 by 125. $137 \div 125 = 1 \text{ with a remainder of } 12$ So, we use one Circle symbol. We have 12 remaining. Now we look at 12. The largest power of 5 less than or equal to 12 is $5^1 = 5$. We divide 12 by 5. $12 \div 5 = 2 \text{ with a remainder of } 2$ So, we use two Square symbols. We have 2 remaining. Now we look at 2. The largest power of 5 less than or equal to 2 is $5^0 = 1$. We divide 2 by 1. $2 \div 1 = 2 \text{ with a remainder of } 0$ So, we use two Triangle symbols. Combining these, we get: $137 = 1 \times 125 + 2 \times 5 + 2 \times 1$

$\boxed{137 = \bigcirc \square \square \triangle \triangle}$

Step 5 — Converting 293

We want to find how many of each power of 5 are in 293. The largest power of 5 less than or equal to 293 is $5^3 = 125$. We divide 293 by 125. $293 \div 125 = 2 \text{ with a remainder of } 43$ So, we use two Circle symbols. We have 43 remaining. Now we look at 43. The largest power of 5 less than or equal to 43 is $5^2 = 25$. We divide 43 by 25. $43 \div 25 = 1 \text{ with a remainder of } 18$ So, we use one Hexagon symbol. We have 18 remaining. Now we look at 18. The largest power of 5 less than or equal to 18 is $5^1 = 5$. We divide 18 by 5. $18 \div 5 = 3 \text{ with a remainder of } 3$ So, we use three Square symbols. We have 3 remaining. Now we look at 3. The largest power of 5 less than or equal to 3 is $5^0 = 1$. We divide 3 by 1. $3 \div 1 = 3 \text{ with a remainder of } 0$ So, we use three Triangle symbols. Combining these, we get: $293 = 2 \times 125 + 1 \times 25 + 3 \times 5 + 3 \times 1$

$\boxed{293 = \bigcirc \bigcirc \hexagon \square \square \square \triangle \triangle \triangle}$

Step 6 — Converting 651

We want to find how many of each power of 5 are in 651. The largest power of 5 less than or equal to 651 is $5^4 = 625$. We divide 651 by 625. $651 \div 625 = 1 \text{ with a remainder of } 26$ So, we use one Tilde symbol. We have 26 remaining. Now we look at 26. The largest power of 5 less than or equal to 26 is $5^2 = 25$. We divide 26 by 25. $26 \div 25 = 1 \text{ with a remainder of } 1$ So, we use one Hexagon symbol. We have 1 remaining. Now we look at 1. The largest power of 5 less than or equal to 1 is $5^0 = 1$. We divide 1 by 1. $1 \div 1 = 1 \text{ with a remainder of } 0$ So, we use one Triangle symbol. Combining these, we get: $651 = 1 \times 625 + 1 \times 25 + 1 \times 1$

$\boxed{651 = \sim \hexagon \triangle}$

Answer

(i) $15 = \square \square \square$ (ii) $50 = \hexagon \hexagon$ (iii) $137 = \bigcirc \square \square \triangle \triangle$ (iv) $293 = \bigcirc \bigcirc \hexagon \square \square \square \triangle \triangle \triangle$ (v) $651 = \sim \hexagon \triangle$