Electrician Apprentice Aptitude Test

A variable is a letter or symbol representing an unknown quantity. Common variables include x, y, n, and a. Variables allow us to write general mathematical statements and solve for unknown values.

Terms are the parts of an expression separated by addition or subtraction. In the expression 3x plus 2y minus 5, the terms are 3x, 2y, and negative 5. The numerical part of a term is the coefficient. In 3x, the coefficient is 3.

Like terms have the same variable raised to the same power. In 5x plus 3y minus 2x plus y, the terms 5x and negative 2x are like terms, as are 3y and y. Constants are like terms with each other.

Combining like terms simplifies expressions. Add or subtract the coefficients while keeping the variable unchanged. For 5x minus 2x, subtract coefficients to get 3x. For 3y plus y, add coefficients to get 4y.

An equation states that two expressions are equal. The equal sign separates the two sides. Examples include x plus 5 equals 12, 3y minus 7 equals 14, and 2(n plus 3) equals 16.

Translating words to equations requires identifying the unknown and the relationships. "A number increased by 5 equals 12" translates to x plus 5 equals 12. "Three times a number minus 7 equals 14" translates to 3y minus 7 equals 14. "Twice the sum of a number and 3 equals 16" translates to 2(n plus 3) equals 16.

Key phrases for addition include "sum," "plus," "increased by," "more than," and "added to." Key phrases for subtraction include "difference," "minus," "decreased by," "less than," and "subtracted from." Key phrases for multiplication include "product," "times," "of," and "multiplied by." Key phrases for division include "quotient," "divided by," "per," and "ratio."

Solving equations relies on properties that maintain equality while isolating the variable.

The addition property of equality states that adding the same number to both sides maintains equality. If a equals b, then a plus c equals b plus c. This allows us to eliminate subtraction by adding.

The subtraction property states that subtracting the same number from both sides maintains equality. If a equals b, then a minus c equals b minus c. This allows us to eliminate addition by subtracting.

The multiplication property states that multiplying both sides by the same nonzero number maintains equality. If a equals b, then a times c equals b times c. This allows us to eliminate division by multiplying.

The division property states that dividing both sides by the same nonzero number maintains equality. If a equals b, then a divided by c equals b divided by c. This allows us to eliminate multiplication by dividing.

One-step equations require a single operation to isolate the variable.

For addition equations like x plus 7 equals 15, subtract 7 from both sides. This gives x equals 8. Check by substituting: 8 plus 7 equals 15 is true.

For subtraction equations like y minus 4 equals 9, add 4 to both sides. This gives y equals 13. Check: 13 minus 4 equals 9 is true.

For multiplication equations like 5n equals 35, divide both sides by 5. This gives n equals 7. Check: 5 times 7 equals 35 is true.

For division equations like a divided by 3 equals 6, multiply both sides by 3. This gives a equals 18. Check: 18 divided by 3 equals 6 is true.

Two-step equations require two operations. The general strategy is to undo addition or subtraction first, then undo multiplication or division.

Solve 3x plus 5 equals 20. First, subtract 5 from both sides to get 3x equals 15. Then divide both sides by 3 to get x equals 5. Check: 3 times 5 plus 5 equals 15 plus 5 equals 20 is true.

Solve 2y minus 7 equals 11. First, add 7 to both sides to get 2y equals 18. Then divide both sides by 2 to get y equals 9. Check: 2 times 9 minus 7 equals 18 minus 7 equals 11 is true.

Solve n divided by 4 plus 3 equals 7. First, subtract 3 from both sides to get n divided by 4 equals 4. Then multiply both sides by 4 to get n equals 16. Check: 16 divided by 4 plus 3 equals 4 plus 3 equals 7 is true.

Multi-step equations may require simplifying one or both sides before solving.

Solve 2(x plus 3) equals 14. First, distribute the 2 to get 2x plus 6 equals 14. Subtract 6 from both sides to get 2x equals 8. Divide by 2 to get x equals 4.

Solve 5x minus 2x plus 4 equals 16. First, combine like terms on the left to get 3x plus 4 equals 16. Subtract 4 to get 3x equals 12. Divide by 3 to get x equals 4.

Solve 3(2y minus 1) plus 4 equals 19. Distribute to get 6y minus 3 plus 4 equals 19. Combine constants to get 6y plus 1 equals 19. Subtract 1 to get 6y equals 18. Divide by 6 to get y equals 3.

When variables appear on both sides, move all variable terms to one side and all constants to the other.

Solve 5x plus 3 equals 2x plus 15. Subtract 2x from both sides to get 3x plus 3 equals 15. Subtract 3 from both sides to get 3x equals 12. Divide by 3 to get x equals 4. Check: 5 times 4 plus 3 equals 23, and 2 times 4 plus 15 equals 23 is true.

Solve 7y minus 8 equals 3y plus 12. Subtract 3y from both sides to get 4y minus 8 equals 12. Add 8 to both sides to get 4y equals 20. Divide by 4 to get y equals 5.

Solve 2(n minus 4) equals 3(n plus 1). Distribute to get 2n minus 8 equals 3n plus 3. Subtract 2n from both sides to get negative 8 equals n plus 3. Subtract 3 from both sides to get negative 11 equals n, or n equals negative 11.

Some equations have no solution. When simplification leads to a false statement like 5 equals 3, the equation has no solution. Example: Solve 2x plus 5 equals 2x minus 3. Subtract 2x from both sides to get 5 equals negative 3, which is false. No value of x satisfies this equation.

Some equations have infinitely many solutions. When simplification leads to a true statement like 4 equals 4, every real number is a solution. Example: Solve 3(x plus 2) equals 3x plus 6. Distribute to get 3x plus 6 equals 3x plus 6. This is always true, so all real numbers are solutions.

Equations containing fractions can be solved by clearing fractions first. Multiply every term by the least common denominator of all fractions.

Solve x divided by 2 plus x divided by 3 equals 5. The LCD of 2 and 3 is 6. Multiply every term by 6: 6 times x divided by 2 plus 6 times x divided by 3 equals 6 times 5. This gives 3x plus 2x equals 30, so 5x equals 30, and x equals 6.

Solve (2x minus 1) divided by 4 equals 3. Multiply both sides by 4 to get 2x minus 1 equals 12. Add 1 to get 2x equals 13. Divide by 2 to get x equals 6.5.

Solve (x plus 2) divided by 3 minus (x minus 1) divided by 2 equals 1. The LCD is 6. Multiply every term by 6: 2(x plus 2) minus 3(x minus 1) equals 6. Distribute: 2x plus 4 minus 3x plus 3 equals 6. Combine: negative x plus 7 equals 6. Subtract 7: negative x equals negative 1. Multiply by negative 1: x equals 1.

Equations with decimals can be solved directly or by clearing decimals first.

Solve 0.5x plus 1.2 equals 3.7 directly. Subtract 1.2 from both sides to get 0.5x equals 2.5. Divide by 0.5 to get x equals 5.

Alternatively, clear decimals by multiplying by a power of 10. For 0.5x plus 1.2 equals 3.7, multiply every term by 10 to get 5x plus 12 equals 37. Subtract 12 to get 5x equals 25. Divide by 5 to get x equals 5.

Solve 0.03y minus 0.5 equals 0.1. Multiply by 100 to get 3y minus 50 equals 10. Add 50 to get 3y equals 60. Divide by 3 to get y equals 20.

A formula expresses a relationship between quantities. Common formulas include A equals lw for area of a rectangle, d equals rt for distance, and C equals 2 times pi times r for circumference.

Sometimes you need to solve a formula for a different variable. Solve A equals lw for w. Divide both sides by l to get w equals A divided by l.

Solve d equals rt for t. Divide both sides by r to get t equals d divided by r.

Solve 2x plus 3y equals 12 for y. Subtract 2x from both sides to get 3y equals 12 minus 2x. Divide by 3 to get y equals (12 minus 2x) divided by 3, which can be written as y equals 4 minus (2/3)x.

Solve the perimeter formula P equals 2l plus 2w for l. Subtract 2w from both sides to get P minus 2w equals 2l. Divide by 2 to get l equals (P minus 2w) divided by 2.

Setting up equations from word problems requires careful reading and identification of the unknown.

A number increased by 7 equals three times the number minus 5. Find the number. Let x be the number. Write the equation: x plus 7 equals 3x minus 5. Solve: subtract x from both sides to get 7 equals 2x minus 5. Add 5 to get 12 equals 2x. Divide by 2 to get x equals 6. The number is 6.

The sum of two consecutive integers is 47. Find the integers. Let n be the first integer. The next consecutive integer is n plus 1. Write the equation: n plus n plus 1 equals 47. Simplify: 2n plus 1 equals 47. Subtract 1: 2n equals 46. Divide by 2: n equals 23. The integers are 23 and 24.

A rectangle has a perimeter of 56 inches. The length is 4 inches more than the width. Find the dimensions. Let w be the width. Then the length is w plus 4. The perimeter formula gives: 2(w plus 4) plus 2w equals 56. Distribute: 2w plus 8 plus 2w equals 56. Combine: 4w plus 8 equals 56. Subtract 8: 4w equals 48. Divide by 4: w equals 12. The width is 12 inches and the length is 16 inches.