Electrician Apprentice Aptitude Test

A fraction represents a part of a whole. It consists of two numbers separated by a horizontal or diagonal line. The top number is the numerator, indicating how many parts you have. The bottom number is the denominator, indicating how many equal parts make up the whole.

When you see 3/4, this means you have 3 parts out of 4 equal parts. The denominator tells you the size of each part, while the numerator tells you how many of those parts you possess.

Proper fractions have numerators smaller than denominators. Examples include 1/2, 3/4, and 5/8. These fractions represent values less than one whole. Improper fractions have numerators equal to or larger than denominators. Examples include 5/3, 7/4, and 9/2. These represent values equal to or greater than one whole.

Mixed numbers combine a whole number with a proper fraction. The number 2 3/4 means two whole units plus three-quarters of another unit. To convert a mixed number to an improper fraction, multiply the whole number by the denominator, add the numerator, and place this sum over the original denominator. For 2 3/4, calculate 2 times 4 equals 8, add 3 to get 11, yielding 11/4.

To convert an improper fraction to a mixed number, divide the numerator by the denominator. The quotient becomes the whole number, and the remainder becomes the new numerator over the original denominator. For 17/5, divide 17 by 5 to get 3 with remainder 2, yielding 3 2/5.

Equivalent fractions represent the same value using different numbers. The fractions 1/2, 2/4, 3/6, and 4/8 all represent the same quantity. You create equivalent fractions by multiplying or dividing both numerator and denominator by the same nonzero number.

To find an equivalent fraction, multiply both parts by your chosen number. Starting with 2/3 and multiplying both by 4 gives 8/12. To simplify a fraction, divide both parts by their greatest common factor. Starting with 12/18 and dividing both by 6 gives 2/3.

The greatest common factor of two numbers is the largest number that divides evenly into both. For 12 and 18, the factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 18 are 1, 2, 3, 6, 9, and 18. The greatest common factor is 6.

A fraction is in lowest terms or simplest form when the numerator and denominator share no common factors other than 1. The fraction 3/4 is in lowest terms because 3 and 4 share no common factors. The fraction 6/8 is not in lowest terms because both are divisible by 2.

When fractions share the same denominator, add or subtract the numerators while keeping the denominator unchanged. For 3/8 plus 2/8, add the numerators to get 5/8. For 7/10 minus 3/10, subtract to get 4/10, which simplifies to 2/5.

When fractions have different denominators, you must find a common denominator before operating. The least common denominator is the smallest number that both denominators divide into evenly. For 1/3 and 1/4, the least common denominator is 12 because 12 is the smallest number divisible by both 3 and 4.

To find the least common denominator, list multiples of each denominator until you find a match. Multiples of 3 are 3, 6, 9, 12, 15. Multiples of 4 are 4, 8, 12, 16. The least common multiple is 12.

Convert each fraction to an equivalent fraction with the common denominator. For 1/3, multiply both parts by 4 to get 4/12. For 1/4, multiply both parts by 3 to get 3/12. Now add: 4/12 plus 3/12 equals 7/12.

When adding mixed numbers, you can add the whole numbers separately from the fractions. For 2 1/4 plus 3 2/3, first find a common denominator of 12. Convert to 2 3/12 plus 3 8/12. Add whole numbers to get 5, add fractions to get 11/12, yielding 5 11/12.

Multiplication of fractions does not require common denominators. Multiply the numerators together to get the new numerator. Multiply the denominators together to get the new denominator. For 2/3 times 4/5, calculate 2 times 4 equals 8 and 3 times 5 equals 15, yielding 8/15.

You can simplify before multiplying by canceling common factors. For 3/4 times 8/9, notice that 3 and 9 share factor 3, and 4 and 8 share factor 4. Cancel to get 1/1 times 2/3, which equals 2/3.

To multiply mixed numbers, first convert them to improper fractions. For 2 1/2 times 1 1/3, convert to 5/2 times 4/3. Multiply to get 20/6, which simplifies to 10/3 or 3 1/3.

To multiply a fraction by a whole number, write the whole number as a fraction with denominator 1. For 3/4 times 6, write as 3/4 times 6/1, yielding 18/4 or 4 1/2.

Division of fractions uses the reciprocal method. The reciprocal of a fraction flips the numerator and denominator. The reciprocal of 2/3 is 3/2. The reciprocal of 5 is 1/5.

To divide fractions, multiply the first fraction by the reciprocal of the second. For 3/4 divided by 2/5, multiply 3/4 by 5/2 to get 15/8 or 1 7/8.

To divide mixed numbers, convert to improper fractions first. For 3 1/2 divided by 1 1/4, convert to 7/2 divided by 5/4. Multiply 7/2 by 4/5 to get 28/10 or 2 4/5.

Decimals represent parts of whole numbers using place value extending to the right of the decimal point. Each position represents a power of ten. The first place after the decimal is tenths, the second is hundredths, the third is thousandths, and so on.

The number 3.472 means 3 ones, 4 tenths, 7 hundredths, and 2 thousandths. Reading decimals uses the last place value as the denominator. Read 3.472 as "three and four hundred seventy-two thousandths."

Comparing decimals requires aligning decimal points and comparing place by place from left to right. To compare 0.45 and 0.449, align them as 0.450 and 0.449. Compare: both have 0 ones, both have 4 tenths, both have 4 hundredths in the second number but 5 hundredths in the first. Since 5 hundredths exceeds 4 hundredths, 0.45 is greater.

To convert a fraction to a decimal, divide the numerator by the denominator. For 3/4, divide 3 by 4 to get 0.75. For 1/3, divide 1 by 3 to get 0.333... (a repeating decimal).

Some fractions produce terminating decimals that end. Examples include 1/2 equals 0.5, 1/4 equals 0.25, and 1/8 equals 0.125. Other fractions produce repeating decimals that continue infinitely with a pattern. Examples include 1/3 equals 0.333..., 1/6 equals 0.1666..., and 1/7 equals 0.142857142857...

To convert a terminating decimal to a fraction, write the decimal digits as the numerator and use the place value as the denominator. For 0.75, write 75/100, which simplifies to 3/4. For 0.125, write 125/1000, which simplifies to 1/8.

To convert a repeating decimal to a fraction requires algebra. For 0.333..., let x equal 0.333.... Then 10x equals 3.333.... Subtract: 10x minus x equals 3.333... minus 0.333..., giving 9x equals 3, so x equals 3/9 or 1/3.

Adding and subtracting decimals requires aligning decimal points vertically. Fill empty places with zeros as needed. For 3.45 plus 2.8, align as 3.45 plus 2.80 to get 6.25. For 5.2 minus 3.75, align as 5.20 minus 3.75 to get 1.45.

Multiplying decimals does not require alignment. Multiply as with whole numbers, ignoring the decimal points temporarily. Count the total decimal places in both factors. Place the decimal point in the product so it has that many decimal places. For 2.3 times 1.5, multiply 23 times 15 to get 345. The factors have one decimal place each, totaling two. Place the decimal to get 3.45.

Dividing decimals requires making the divisor a whole number. Move the decimal point in the divisor to the right until it becomes a whole number. Move the decimal point in the dividend the same number of places. For 4.5 divided by 0.15, move both decimal points two places right to get 450 divided by 15, which equals 30.

Percent means "per hundred." A percentage represents a number as parts per 100. The symbol % denotes percent. So 45% means 45 per 100 or 45/100.

Percentages connect directly to fractions and decimals. To convert a percentage to a decimal, divide by 100 or move the decimal point two places left. So 45% becomes 0.45, and 7% becomes 0.07.

To convert a decimal to a percentage, multiply by 100 or move the decimal point two places right. So 0.45 becomes 45%, and 0.07 becomes 7%.

To convert a percentage to a fraction, write the percentage over 100 and simplify. So 45% becomes 45/100, which simplifies to 9/20. For percentages with decimals, clear the decimal first. So 12.5% becomes 12.5/100, multiply both by 10 to get 125/1000, which simplifies to 1/8.

To convert a fraction to a percentage, divide to get a decimal, then multiply by 100. For 3/8, divide 3 by 8 to get 0.375, then multiply by 100 to get 37.5%.

Finding a percentage of a number uses multiplication. Convert the percentage to a decimal and multiply. For 25% of 80, calculate 0.25 times 80 to get 20.

The word "of" in percentage problems indicates multiplication. "What is 15% of 60?" means calculate 0.15 times 60, which equals 9.

Finding what percentage one number is of another uses division. Divide the part by the whole and multiply by 100. If 15 is what percentage of 60? Calculate 15 divided by 60 to get 0.25, then multiply by 100 to get 25%.

Finding the whole when given a percentage uses division. If 30 is 25% of what number? Set up the equation: 30 equals 0.25 times the unknown. Divide 30 by 0.25 to get 120.

Percentage increase calculates how much a value has grown relative to its original size. The formula is: percentage increase equals the amount of increase divided by the original amount, times 100.

If a price rises from 50 to 65, the increase is 15. The percentage increase is 15 divided by 50, times 100, which equals 30%.

Percentage decrease works similarly. If a price drops from 80 to 60, the decrease is 20. The percentage decrease is 20 divided by 80, times 100, which equals 25%.

To increase a number by a percentage, multiply by (1 plus the decimal form of the percentage). To increase 200 by 15%, multiply 200 by 1.15 to get 230.

To decrease a number by a percentage, multiply by (1 minus the decimal form of the percentage). To decrease 200 by 15%, multiply 200 by 0.85 to get 170.

Memorizing common equivalents speeds up calculations. One half equals 0.5 equals 50%. One third equals approximately 0.333 equals approximately 33.33%. One fourth equals 0.25 equals 25%. One fifth equals 0.2 equals 20%. One eighth equals 0.125 equals 12.5%. Three fourths equals 0.75 equals 75%. Two thirds equals approximately 0.667 equals approximately 66.67%.

Word problems require translating English into mathematical operations. Key phrases indicate operations. "Of" usually means multiply. "Is" usually means equals. "What" indicates the unknown. "Per" indicates division.

A recipe calls for 2 1/2 cups of flour. You want to make 1 1/2 times the recipe. How much flour do you need? Multiply 2 1/2 by 1 1/2. Convert to improper fractions: 5/2 times 3/2 equals 15/4 or 3 3/4 cups.

A store offers 20% off a 75 dollar item. What is the sale price? Calculate the discount: 0.20 times 75 equals 15. Subtract from original: 75 minus 15 equals 60 dollars. Alternatively, multiply 75 by 0.80 to get 60.

A worker earns 18.50 dollars per hour and works 37.5 hours. What is the gross pay? Multiply 18.50 by 37.5 to get 693.75 dollars.