Solving 2-Digit Number Word Problems

Let's break down this word problem step-by-step to find the original 2-digit number. We are given two conditions about the digits and the number itself. We will use algebra to represent the unknown digits and the number, and then solve the resulting equations.

Representing the 2-Digit Number

A 2-digit number can be represented based on its tens digit and unit digit. Let:

• The unit digit be represented by the variable 'u'.
• The tens digit be represented by the variable 't'.

The value of the original 2-digit number is $10 \times (\text{tens digit}) + (\text{unit digit})$, which can be written as $\small{10t + u}$.

When the digits are interchanged, the new number has 'u' in the tens place and 't' in the unit place. The value of the number obtained by interchanging the digits is $\small{10u + t}$.

Translating Conditions into Equations

The problem gives us two conditions:

1. Condition 1: The tens digit is two times its unit digit.
2. Condition 2: The number is 12 less than two times the number obtained by interchanging its digits.

Let's write these conditions as algebraic equations:

• From Condition 1: $\small{t = 2u}$
• From Condition 2: $\small{10t + u = 2(10u + t) - 12}$

Now we have a system of two linear equations with two variables ('t' and 'u'):

Equation 1: $\small{t = 2u}$
Equation 2: $\small{10t + u = 20u + 2t - 12}$

Solving the System of Equations

We can use the substitution method to solve these equations. Substitute the value of 't' from Equation 1 into Equation 2:

Substitute $\small{t = 2u}$ into $\small{10t + u = 20u + 2t - 12}$:

$\small{10(2u) + u = 20u + 2(2u) - 12}$

Simplify the equation:

$\small{20u + u = 20u + 4u - 12}$

$\small{21u = 24u - 12}$

Now, rearrange the terms to solve for 'u'. Subtract $\small{21u}$ from both sides:

$\small{0 = 24u - 21u - 12}$

$\small{0 = 3u - 12}$

Add 12 to both sides:

$\small{12 = 3u}$

Divide by 3 to find 'u':

$\small{u = \frac{12}{3}}$

$\small{u = 4}$

Now that we have the value of the unit digit 'u', we can find the value of the tens digit 't' using Equation 1 ($\small{t = 2u}$):

$\small{t = 2 \times 4}$

$\small{t = 8}$

Finding the Original Number

The unit digit is 4 and the tens digit is 8. The original 2-digit number is $\small{10t + u}$.

Original Number = $\small{10(8) + 4}$
Original Number = $\small{80 + 4}$
Original Number = $\small{84}$

Verification

Let's check if the number 84 satisfies both original conditions:

• Condition 1: Tens digit is two times the unit digit.

  Tens digit = 8, Unit digit = 4. Is $8 = 2 \times 4$? Yes, $8=8$. This condition is met.

• Condition 2: The number is 12 less than two times the number obtained by interchanging its digits.

  Original number = 84.
  Interchanged digits number = 48.
  Two times the interchanged number = $\small{2 \times 48 = 96}$.
  Is the original number (84) equal to 12 less than 96? $\small{96 - 12 = 84}$. Yes, $84=84$. This condition is also met.

Since both conditions are satisfied, the original number is 84.

Revision Table: Key Concepts for 2-Digit Number Problems

Additional Information: Word Problems and Algebra

Word problems are a common type of question in mathematics that require you to translate a real-world or descriptive scenario into mathematical equations. For problems involving unknown numbers or quantities, algebra is a powerful tool.

Steps to Solve Algebraic Word Problems:

1. Read Carefully: Understand the problem and what you need to find. Identify the unknown quantities.
2. Define Variables: Assign letters (like x, y, u, t) to represent the unknown quantities. Be specific about what each variable represents.
3. Translate to Equations: Convert the sentences describing relationships between quantities into algebraic equations. Look for keywords like "is" (equals), "sum" (addition), "difference" (subtraction), "product" (multiplication), "quotient" (division), "less than" (subtraction), "more than" (addition), "times" (multiplication).
4. Solve the Equations: Use algebraic techniques (substitution, elimination, etc.) to find the values of the variables.
5. Answer the Question: Make sure you provide the answer that the question asks for. Sometimes you find the variable values, but need to calculate something else based on them (like the original number, as in this case).
6. Check Your Answer: Plug the values you found back into the original word problem conditions to ensure they make sense and satisfy all requirements.

Number-based word problems often involve understanding place value (like in 2-digit numbers) or relationships between consecutive integers, fractions, ratios, etc. Practice is key to becoming proficient in translating word problems into solvable mathematical forms.