Which two numbers should be interchanged to make the given equation correct? 75 ÷ 5 + 15 × 4 - 17 × 3 + (60 ÷ 6) × 10 = 100 (Note: Interchange should be done of the entire number and not individual digits of a given number.)
5 and 3
Numbers Interchange to Correct Equation
The core task is to find a pair of numbers within the given mathematical expression that, when swapped, result in the equation holding true. The equation presented is:
\(75 \div 5 + 15 \times 4 - 17 \times 3 + (60 \div 6) \times 10 = 100\)
The instructions clarify that we must swap the entire numerical values, not individual digits.
Equation Evaluation: Original State
First, let's verify the accuracy of the original equation by evaluating it step-by-step, following the standard order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Division and Multiplication (from left to right), Addition and Subtraction (from left to right).
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Parentheses: Evaluate the expression within the parentheses.
\((60 \div 6) = 10\)
The equation transforms to: \(75 \div 5 + 15 \times 4 - 17 \times 3 + 10 \times 10 = 100\)
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Division and Multiplication: Perform these operations from left to right.
- \(75 \div 5 = 15\)
- \(15 \times 4 = 60\)
- \(17 \times 3 = 51\)
- \(10 \times 10 = 100\)
The equation simplifies to: \(15 + 60 - 51 + 100 = 100\)
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Addition and Subtraction: Complete these operations from left to right.
- \(15 + 60 = 75\)
- \(75 - 51 = 24\)
- \(24 + 100 = 124\)
The calculation yields \(124\), which is not equal to the stated result of \(100\). This confirms the original equation is incorrect.
Interchange Options Tested
We will now examine the provided options to find the pair of numbers whose interchange corrects the equation. The options are:
- 3 and 6
- 17 and 15
- 3 and 4
- 5 and 3
Let's test the fourth option: interchanging the numbers 5 and 3.
Correct Equation Achieved: Interchanging 5 and 3
According to option 4, we should swap the values 5 and 3 wherever they appear in the equation.
Original Equation: \(75 \div 5 + 15 \times 4 - 17 \times 3 + (60 \div 6) \times 10 = 100\)
Applying the interchange (5 becomes 3, and 3 becomes 5):
- The term \(75 \div 5\) becomes \(75 \div 3\).
- The term \(17 \times 3\) becomes \(17 \times 5\).
The modified equation is:
\(75 \div 3 + 15 \times 4 - 17 \times 5 + (60 \div 6) \times 10 = 100\)
Equation Evaluation: Corrected State
Now, we evaluate the modified equation using the order of operations (PEMDAS/BODMAS):
-
Parentheses:
\((60 \div 6) = 10\)
The equation is now: \(75 \div 3 + 15 \times 4 - 17 \times 5 + 10 \times 10 = 100\)
-
Division and Multiplication:
- \(75 \div 3 = 25\)
- \(15 \times 4 = 60\)
- \(17 \times 5 = 85\)
- \(10 \times 10 = 100\)
The equation simplifies to: \(25 + 60 - 85 + 100 = 100\)
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Addition and Subtraction:
- \(25 + 60 = 85\)
- \(85 - 85 = 0\)
- \(0 + 100 = 100\)
The final result is \(100\), which correctly matches the right side of the equation. This confirms that interchanging the numbers 5 and 3 is the correct operation to make the equation true.
Similar Questions
Select the set, from the given options, in which the numbers are related in the same way as are the numbers of the following sets.
(NOTE: Operations should be performed on the whole numbers, without breaking down the numbers into its constituent digits. E.g. 13 – Operations on 13 such as adding/subtracting/multiplying etc. to 13 can be performed. Breaking down 13 into 1 and 3 and then performing mathematical operations on 1 and 3 is not allowed.)
(6, 13, 78)
(5, 22, 110)
If + means -, - means ×, × means ÷, ÷ means +, then what will come in place of the question mark (?) in the following equation?
28 - 6 ÷ 4 + 14 × 14 = ?
In the following number-pairs, the second number is obtained by applying certain mathematical operations to the first number. Which numbers should replace X and Y so that the pattern followed by the two numbers on the left side of :: is the same as that on the right side of :: ?
(NOTE: Operations should be performed on the whole numbers, without breaking down the numbers into its constituent digits. E.g. 13 Operations on 13 such as adding/subtracting /multiplying etc. to 13 can be performed. Breaking down 13 into 1 and 3 and then performing mathematical operations on 1 and 3 is not allowed.)
X : 649 :: 956 : Y
If ‘A’ stands for ‘\(\div \)', ‘B’ stands for ‘x', ‘C’ stands for ‘+', ‘D’ stands for ‘-', what will come in the place of the question mark '?' in the following equation?
225 B 15 A 15 C 125 D 225 = ?Select the set in which the numbers are related in the same way as are the numbers of the following sets.
(NOTE : Operations should be performed on the whole numbers, without breaking down the numbers into its constituent digits. E.g. 3 - Operations on 3 such as adding /deleting /multiplying etc. to 3 can be performed, Breaking down 3 into | and 3 and then performing mathematical operations on 1 and 3 is not allowed.)
(144, 12, 6)
(210, 15, 7)Select the set in which the numbers are related in the same way as are the numbers of the following sets.
(NOTE : Operations should be performed on the whole numbers, without breaking down the numbers into its constituent digits. E.g. 3 - Operations on 3 such as adding /deleting /multiplying etc. to 3 can be performed, Breaking down 3 into | and 3 and then performing mathematical operations on | and 3 is not allowed.)
(9, 6, 56)
(7,8, 58)If A means +, B means -, C means x, D means \(\div \), then what will come in place of the question mark (?) in the following equation?
18 D 2 A 4 C 5 B 15 = ?If 'A' stands for '+', 'B' stands for '×', 'C' stands for '+' and 'D' stands for '-', then what will come in place of the question mark (?) in the following equation?
47 D 25 A 5 C 2 B 9 = ?
If + means -, - means ×, × means ÷, ÷ means +, then what will come in place of the question mark (?) in the following equation?
22 - 5 ÷ 3 + 12 × 3 = ?
Which two numbers should be interchanged to make the given equation correct?
186 - 82 + (129 ÷ 3) − 6 × 8 + (31 - 19) × 2 = 85
(Note: Interchange should be done of entire number and not individual digits of a given number.)
Important Questions from Mathematical Operations
Which two numbers (not digits) should be interchanged to make the given equation correct?
21 × 8 – 18 + (64 ÷ 2) + 32 = 64
The two expressions on either side of the equal sign will have the same value if two numbers on the left-hand side are interchanged and two signs on the right-hand side are interchanged. What are those two numbers (not digits) and the two signs?
9 × 1 ÷ 3 + 8 – 7 = 12 ÷ 3 + 2 × 33 – 13
Select the set, from the given options, in which the numbers are related in the same way as are the numbers of the following sets.
(NOTE: Operations should be performed on the whole numbers, without breaking down the numbers into its constituent digits. E.g. 13 – Operations on 13 such as adding/subtracting/multiplying etc. to 13 can be performed. Breaking down 13 into 1 and 3 and then performing mathematical operations on 1 and 3 is not allowed.)
(6, 13, 78)
(5, 22, 110)
If + means -, - means ×, × means ÷, ÷ means +, then what will come in place of the question mark (?) in the following equation?
28 - 6 ÷ 4 + 14 × 14 = ?
In the following number-pairs, the second number is obtained by applying certain mathematical operations to the first number. Which numbers should replace X and Y so that the pattern followed by the two numbers on the left side of :: is the same as that on the right side of :: ?
(NOTE: Operations should be performed on the whole numbers, without breaking down the numbers into its constituent digits. E.g. 13 Operations on 13 such as adding/subtracting /multiplying etc. to 13 can be performed. Breaking down 13 into 1 and 3 and then performing mathematical operations on 1 and 3 is not allowed.)
X : 649 :: 956 : Y
