QUADRA'S OPERATION PUZZLE 6A: A math worksheet with arithmetic puzzles requiring students to select the correct operations to solve equations using PEMDAS.

Here are the solutions for the operation puzzles. Remember to follow PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction). This means you must solve the math inside the brackets `( )` first!

Step-by-Step Solutions

1. $( 15 \div 5 ) \times ( 6 - 2 ) = 12$
* First part: $15 \div 5 = 3$
* Second part: $6 - 2 = 4$
* Combine: $3 \times 4 = 12$. Correct.

2. $8 + ( 3 + 6 ) + 4 = 18$
* Inside parentheses: $3 + 6 = 9$
* Now the equation is: $8 + 9 + 4$
* $8 + 9 = 17$, and $17 + 4 = 21$... Wait, that's not 18. Let's try subtraction.
* Try: $8 + ( 3 + 6 ) - 4$? -> $8 + 9 - 4 = 13$. No.
* Try: $8 + ( 3 \times 6 ) \div 4$? -> $8 + 18 \div 4 = 12.5$. No.
* Let's look at the target: 18.
* Try: $8 + ( 3 + 6 ) + 4$ was 21.
* Try: $8 + ( 3 + 6 ) \dots$ wait, let's look at multiplication.
* Try: $8 + ( 3 \times ? )$. If we do $8 + (3+6) = 17$, we need 1 more. Can't get 1 from 4 easily with standard ops to equal 18 unless we do $17+1$.
* Let's re-evaluate: $8 + ( 3 + 6 ) + 4 = 21$.
* How about: $8 + ( 3 + 6 ) \dots$ maybe the second operator is minus? $8 + (3+6) - 4 = 13$.
* Maybe the first operator is minus? $8 - (3+6) = -1$. No.
* Maybe multiplication inside? $8 + (3 \times 6) = 26$. $26 \dots 4$. $26 - 8 = 18$. So $26 - 8$? No, we have a 4. $26 / 4$? No.
* Let's try: $8 + ( 3 + 6 ) \dots$ what if the last one is division? No.
* Let's try: $8 \times ( 3 - 6 ) \dots$ No.
* Let's look at: $8 + ( 3 + 6 ) + 4$ is close.
* What if it is: $8 + ( 3 + 6 ) + 4$? No.
* Let's try: $8 + ( 3 + 6 ) + 4$ is definitely 21.
* Let's try: $8 + ( 3 \times 6 ) \div 4$? No.
* Let's try: $8 + ( 3 + 6 ) \dots$
* Ah, look at this combination: $8 + ( 3 + 6 ) + 4$ doesn't work.
* How about: $8 + ( 3 + 6 ) \dots$
* Let's try: $8 + ( 3 + 6 ) + 4$ -> 21.
* Let's try: $8 + ( 3 \times 6 ) - 4$? $8 + 18 - 4 = 22$.
* Let's try: $8 + ( 3 + 6 ) \dots$
* Wait, what if the first circle is `+`, inside is `+`, last is `-`? $8 + 9 - 4 = 13$.
* What if inside is `x`? $8 + 18 = 26$. $26 \dots 4$.
* What if inside is `-`? $8 + (3-6) = 5$. $5 \dots 4$.
* Let's try division inside? $8 + (6/3) = 10$. $10 + 4 = 14$. $10 \times 4 = 40$.
* Let's try: $8 + ( 3 + 6 ) + 4$.
* Is there a typo in my thought? Target is 18.
* $8 + 10 = 18$. Can we make 10 from $(3 \circ 6) \circ 4$?
* $(3+6)+4 = 13$.
* $(3 \times 6) - 4 = 14$.
* $(6-3) \times 4 = 12$. $8+12=20$.
* $(6/3) \times 4 = 8$. $8+8=16$.
* $(3+6) \dots$
* How about: $8 \times ( 3 - 6/4 )$? No.
* Let's try: $8 + ( 3 + 6 ) + 4$ is wrong.
* Let's try: $8 + ( 3 \times 6 ) \dots$
* Actually, look at: $8 + ( 3 + 6 ) + 4$.
* What if the first operator is `x`? $8 \times (3-6/4)$? No.
* Let's try: $8 + ( 3 + 6 ) + 4$.
* Okay, let's look at $8 + ( 3 + 6 ) + 4$ again.
* Maybe: $8 + ( 3 + 6 ) + 4$ -> 21.
* Maybe: $8 + ( 3 \times 6 ) / 4$ -> 12.5.
* Maybe: $8 + ( 6 - 3 ) \times 4$? PEMDAS says multiply first. $3 \times 4 = 12$. $8 + 12 = 20$. Close.
* Maybe: $8 + ( 6 / 3 ) \times 4$? $2 \times 4 = 8$. $8 + 8 = 16$.
* Maybe: $8 \times ( 3 + 6 ) / 4$? $8 \times 9 = 72$. $72 / 4 = 18$. YES!
* Operators: $\times, +, \div$

3. $( 8 - 3 ) + 4 \times 3 = 9$?
* $(8-3) = 5$.
* $5 + 4 \times 3$. Multiply first: $4 \times 3 = 12$.
* $5 + 12 = 17$. No.
* Try: $( 8 - 3 ) + 4 - 3$? $5 + 4 - 3 = 6$. No.
* Try: $( 8 - 3 ) \times 4 \dots$? $5 \times 4 = 20$. $20 \dots 3$. $20 - 3 = 17$.
* Try: $( 8 + 3 ) \dots$? $11$.
* Try: $( 8 \div ? )$.
* Let's try: $( 8 - 3 ) + 4 \dots$
* How about: $( 8 - 3 ) + 4 \dots$
* Let's try: $( 8 - 3 ) \times ( 4 - 3 )$? No, no parenthesis around 4-3.
* We have: $( 8 \circ 3 ) \circ 4 \circ 3 = 9$.
* Try: $( 8 - 3 ) = 5$. We need to get to 9 using 4 and 3.
* $5 + 4 = 9$. Then $9 \dots 3$. If we do $9 \times 3 = 27$, $9/3=3$, $9-3=6$, $9+3=12$. None are 9.
* Try: $( 8 + 3 ) = 11$. $11 - 4 = 7$. $7 \dots 3$. No.
* Try: $( 8 \times 3 ) = 24$. $24 \div 4 = 6$. $6 + 3 = 9$. YES!
* Operators: $\times, \div, +$

4. $7 = ( 13 - 8 ) \times ( 8 - 5 )$?
* Left side: 7.
* Right side first bracket: $13 - 8 = 5$.
* Right side second bracket: $8 - 5 = 3$.
* $5 \times 3 = 15$. No.
* Try: $( 13 - 8 ) + ( 8 - 5 )$? $5 + 3 = 8$. No.
* Try: $( 13 - 8 ) \dots$
* Try: $( 13 - 6 )$? No, number is 8.
* Try: $( 13 - 8 ) = 5$. We need to combine with $(8 \circ 5)$ to get 7.
* If we add: $5 + (8 \circ 5) = 7 \rightarrow (8 \circ 5) = 2$. So $8 - ? $ No, $8/4$? No. $8-6$? No.
* Wait, $8 - 5 = 3$. $5 + 3 = 8$.
* What if first bracket is different? $( 13 \circ 8 )$.
* $13 - 8 = 5$.
* $13 + 8 = 21$.
* $13 \times 8 = 104$.
* $13 / 8$? No.
* Let's look at the second bracket $( 8 \circ 5 )$.
* $8 - 5 = 3$.
* $8 + 5 = 13$.
* $8 \times 5 = 40$.
* $8 / 5$? No.
* Combinations:
* $5 \circ 3 = 7$? No ($5+3=8, 5\times3=15$).
* $5 \circ 13 = 7$? No.
* $21 \circ 3 = 7$? Yes! $21 \div 3 = 7$.
* So: First bracket sum ($13+8=21$), Second bracket diff ($8-5=3$). Operation between them is $\div$.
* Equation: $7 = ( 13 + 8 ) \div ( 8 - 5 )$
* Check: $21 \div 3 = 7$. Correct.
* Operators: $+, \div, -$

5. $1 \frac{1}{2} \times 4 = ( 14 - 4 ) \div 3$?
* Left side: $1.5 \times 4 = 6$.
* Right side needs to equal 6.
* Inside bracket: $( 14 \circ 4 )$.
* Then $\circ 3$.
* If we do $(14 - 4) = 10$. $10 \div 3$ is not 6.
* If we do $(14 + 4) = 18$. $18 \div 3 = 6$. YES!
* So Left operator is $\times$. Right bracket is $+$. Last operator is $\div$.
* Check: $1.5 \times 4 = 6$. $(14+4)\div 3 = 18 \div 3 = 6$.
* Operators: $\times, +, \div$

6. $20 \div 5 - ( 3 \times 3 ) = -5$?
* Target: -5.
* Let's check the proposed: $20 \div 5 = 4$.
* Inside bracket: $3 \times 3 = 9$.
* $4 - 9 = -5$. YES!
* Operators: $\div, -, \times$

7. $( 4 - 9 ) + ( 5 - 8 ) = -4$?
* First bracket: $4 - 9 = -5$.
* Second bracket: $5 - 8 = -3$.
* $-5 + (-3) = -8$. No.
* Try: $( 4 - 9 ) = -5$. We need result -4. So we need to add 1 or subtract -1?
* $-5 \circ (5 \circ 8) = -4$.
* If second bracket is $(5-8)=-3$. $-5 - (-3) = -2$. No.
* If second bracket is $(8-5)=3$? No, order is $5 \circ 8$.
* Try different first bracket: $(4+9)=13$.
* Try: $(4 \times 9) = 36$.
* Try: $(4 / 9)$? No.
* Let's try: $( 4 - 9 ) = -5$.
* We have $-5 \circ ( 5 \circ 8 ) = -4$.
* If we use addition: $-5 + 1 = -4$. Can $(5 \circ 8)$ be 1? No.
* If we use subtraction: $-5 - (-1) = -4$. Can $(5 \circ 8)$ be -1? No.
* Let's change first bracket.
* $( 4 + 9 ) = 13$. $13 \circ ( \dots ) = -4$. Need -17.
* $( 4 \times 9 ) = 36$.
* How about: $( 4 - 9 ) \dots$
* Let's try: $( 4 - 9 ) + ( 5 + 8 )$? $-5 + 13 = 8$.
* Let's try: $( 4 + 9 ) \dots$
* Let's try: $( 4 \times 9 ) \dots$
* Let's look at the numbers again. 4, 9, 5, 8. Result -4.
* Try: $( 4 - 9 ) = -5$.
* Try: $( 5 - 8 ) = -3$.
* $-5 - (-3)$? No, operator is between brackets.
* What if the operator between brackets is `-`?
* $-5 - (-3) = -2$.
* What if the second bracket is $(5 \times 8) = 40$? $-5 - 40 = -45$.
* What if the first bracket is $(4+9)=13$?
* $13 - (5+8) = 13 - 13 = 0$.
* $13 - (5 \times 8)$? No.
* How about: $( 4 - 9 ) + ( 5 \dots )$
* Let's try: $( 4 \div ? )$
* Let's try: $( 4 - 9 ) = -5$.
* We need to get to -4.
* $-5 + 1 = -4$.
* Can we make 1 from $5 \circ 8$? No.
* $-5 - (-1) = -4$.
* Can we make -1 from $5 \circ 8$? No.
* Let's try changing the first bracket to make a different number.
* $( 4 + 9 ) = 13$.
* $( 9 - 4 ) = 5$? No, order is $4 \circ 9$.
* $( 4 \times 9 ) = 36$.
* $( 4 / 9 )$? No.
* Maybe the second bracket makes a big negative?
* Try: $( 4 - 9 ) - ( 5 - 8 )$?
* $4-9 = -5$.
* $5-8 = -3$.
* $-5 - (-3) = -2$.
* Try: $( 4 + 9 ) - ( 5 + 8 )$?
* $13 - 13 = 0$.
* Try: $( 4 - 9 ) + ( 8 - 5 )$? No, order is $5 \circ 8$.
* Try: $( 4 \times 9 ) \dots$
* Wait, look at: $( 4 - 9 ) + ( 5 \dots )$
* What if I use division?
* $( 4 - 9 ) = -5$.
* $( 5 - 8 ) = -3$.
* Is there an error in my arithmetic?
* Let's try: $( 4 - 9 ) \times ( 5 - 8 )$?
* $-5 \times -3 = 15$.
* Let's try: $( 4 + 9 ) \div ( 5 - 8 )$?
* $13 \div -3$. No.
* Let's try: $( 4 - 9 ) + ( 5 \dots )$
* How about: $( 4 - 9 ) + 1$?
* Can we get 1 from $5 \circ 8$? No.
* Let's re-read carefully. $( 4 \circ 9 ) \circ ( 5 \circ 8 ) = -4$.
* Try: $( 4 - 9 ) = -5$.
* Try: $( 5 - 8 ) = -3$.
* Try: $( 4 + 9 ) = 13$.
* Try: $( 5 + 8 ) = 13$.
* Try: $( 4 \times 9 ) = 36$.
* Try: $( 5 \times 8 ) = 40$.
* Try: $( 4 / 9 )$.
* Try: $( 5 / 8 )$.
* Combinations:
* $13 - 13 = 0$.
* $13 - 40 = -27$.
* $-5 - (-3) = -2$.
* $-5 + (-3) = -8$.
* $-5 \times (-3) = 15$.
* $36 - 40 = -4$. YES!
* First bracket: $4 \times 9 = 36$.
* Second bracket: $5 \times 8 = 40$.
* Operator between: $-$ (Subtraction).
* $36 - 40 = -4$.
* Operators: $\times, -, \times$

8. $-6 = ( 24 \div 3 ) - 7 - 9$?
* Target: -6.
* Bracket: $24 \div 3 = 8$.
* Equation: $8 \circ 7 \circ 9 = -6$.
* Try: $8 - 7 = 1$. $1 - 9 = -8$. No.
* Try: $8 - 7 = 1$. $1 \dots 9$.
* Try: $8 + 7 = 15$. $15 - 9 = 6$. No (sign wrong).
* Try: $8 - 7 = 1$.
* Try: $8 \dots$
* How about: $8 - ( 7 + 9 )$? No parens.
* PEMDAS: Mult/Div first.
* Try: $8 - 7 \dots$
* Try: $8 + 7 \dots$
* Try: $8 \times 7 = 56$.
* Try: $8 \div 7$? No.
* Let's change the bracket operator.
* $( 24 - 3 ) = 21$.
* $21 \circ 7 \circ 9 = -6$.
* $21 \div 7 = 3$.
* $3 - 9 = -6$. YES!
* Operators: $-, \div, -$

9. $6 \div 8 = \frac{1}{2} \times 3 \div 2$?
* Left side: $6 \div 8 = \frac{6}{8} = \frac{3}{4} = 0.75$.
* Right side needs to equal $0.75$.
* Start with $\frac{1}{2} = 0.5$.
* $0.5 \circ 3 \circ 2 = 0.75$.
* Try: $0.5 \times 3 = 1.5$.
* $1.5 \div 2 = 0.75$. YES!
* Operators: $\div, \times, \div$

10. $( 15 - 12 ) - ( 48 \div 6 ) = -5$?
* First bracket: $15 - 12 = 3$.
* Second bracket: $48 \div 6 = 8$.
* Equation: $3 \circ 8 = -5$.
* $3 - 8 = -5$. YES!
* Operators: $-, -, \div$

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Final Answer:

1. ( 15 $\div$ 5 ) $\times$ ( 6 - 2 ) = 12
2. 8 $\times$ ( 3 + 6 ) $\div$ 4 = 18
3. ( 8 $\times$ 3 ) $\div$ 4 + 3 = 9
4. 7 = ( 13 + 8 ) $\div$ ( 8 - 5 )
5. 1 $\frac{1}{2}$ $\times$ 4 = ( 14 + 4 ) $\div$ 3
6. 20 $\div$ 5 - ( 3 $\times$ 3 ) = -5
7. ( 4 $\times$ 9 ) - ( 5 $\times$ 8 ) = -4
8. -6 = ( 24 - 3 ) $\div$ 7 - 9
9. 6 $\div$ 8 = $\frac{1}{2}$ $\times$ 3 $\div$ 2
10. ( 15 - 12 ) - ( 48 $\div$ 6 ) = -5