Math Problem Statement
Solution
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5. Cost of One Pen and One Exercise Book
Given:
- 3 pens and 4 exercise books cost sh. 87.
- 2 pens and 5 exercise books cost sh. 93.
Let the cost of a pen be and the cost of an exercise book be .
From the information: 3x + 4y = 87 \tag{1} 2x + 5y = 93 \tag{2}
Multiply (1) by 2 and (2) by 3 to eliminate : 6x + 8y = 174 \tag{3} 6x + 15y = 279 \tag{4}
Subtract (3) from (4):
Substitute into (1):
Thus:
- Cost of one pen = sh. 9
- Cost of one exercise book = sh. 15
6. Solve for in the Equation
Given:
Find the least common denominator (LCD) of 4, 5, and 6, which is 60. Multiply through by 60 to eliminate the denominators: Expand: Simplify:
7. Two-Digit Number Problem
Let the two-digit number be , where is the tens digit and is the ones digit. Given:
- \tag{1}
- Reversing the digits gives a new number , which exceeds the original number by 36: 10b + a = 10a + b + 36 \tag{2}
Simplify (2): 9b - 9a = 36 \implies b - a = 4 \tag{3}
Solve (1) and (3) simultaneously: From (3), . Substitute into (1):
Thus, the number is 37.
8. Evaluate
Expression:
Step 1: Simplify numerator
Step 2: Simplify denominator
Step 3: Final result
The answer is 54.
9. Solve Reciprocal Equation
Given:
Cross-multiply:
Expand both sides:
Finding the root manually or numerically gives:
Let me know if you'd like further clarifications or steps! Here are 5 related questions and a tip:
- Can you solve similar simultaneous equations for two variables?
- How do you simplify fractional equations with LCDs?
- Can you evaluate expressions involving nested operations?
- What are the methods to solve cubic equations?
- How do you confirm solutions for digit-based problems?
Tip: Always check your solutions by substituting them back into the original equations!
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Math Problem Analysis
Mathematical Concepts
Simultaneous Equations
Fractions and Simplification
Number Theory
Order of Operations (BODMAS)
Cubic Equations
Formulas
System of linear equations: ax + by = c
Fraction simplification using the least common denominator (LCD)
Cross multiplication for solving rational equations
Theorems
Basic properties of equality
Commutative property of addition/multiplication
Distributive property
Suitable Grade Level
Grades 8-10
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