GED word problems often combine several skills in one question. A reliable approach is to (1) read the question and underline what you are asked to find, (2) pick out the numbers you need, (3) decide the operation (adding or subtracting to combine or compare, multiplying for “of” or “each,” and dividing to split into equal groups or to see how many fit), (4) check whether the answer choices are decimals or fractions so you can choose the more convenient form, and (5) check that your answer is sensible. When a problem asks how many complete or whole items, round down, and for an average, add the values and divide by how many there are.
A decimal is a way to write a part of a whole using place value. The decimal point separates the whole-number part from the part that is less than 1. Moving right from the point, the places are tenths, hundredths, thousandths, and so on, and each one is 10 times smaller than the place before it. When you read a decimal aloud, say the whole part, say “and” for the point, then read the digits as one number followed by the name of the last place. To round, find the place you want, look at the digit just to its right, and round up if it is 5 or more or leave the digit alone if it is 4 or less; then drop everything to the right. To compare decimals, line them up by the decimal point and add zeros at the end so they have the same number of places. Adding zeros at the end of a decimal never changes its value.
Worked Example: (a) Round 6.4172 to the nearest hundredth. (b) Which is greater, 0.35 or 0.305?
(a) The hundredths digit is 1 and the digit to its right is 7, which is 5 or more, so the 1 goes up to 2: 6.42.
(b) Write 0.35 as 0.350. Now compare 350 thousandths with 305 thousandths. Since 350 > 305, 0.35 > 0.305. Notice that having more digits does not make a decimal larger.
1. What digit is in the hundredths place of 28.4617?
Explanation: After the decimal point the places are tenths (4), hundredths (6), thousandths (1) and ten-thousandths (7). The hundredths digit is 6.
2. Round 3.846 to the nearest tenth.
Explanation: The tenths digit is 8 and the digit to its right is 4, which is less than 5. Keep the 8 and drop the rest: 3.8.
3. Which decimal has the greatest value?
Explanation: Write each with three places: 0.710, 0.700, 0.699, 0.070. The greatest is 0.710, so 0.71.
4. Which list shows the numbers in order from least to greatest?
Explanation: With three places: 0.040, 0.400, 0.404, 0.440. In order, that is 0.04, 0.4, 0.404, 0.44.
5. How is “five and thirty-two thousandths” written as a decimal?
Explanation: Thousandths means three decimal places. Thirty-two thousandths is 0.032, so the number is 5.032.
To add or subtract decimals, line up the decimal points so that tenths are under tenths, hundredths under hundredths, and so on. Fill empty places with zeros so every number has the same number of decimal places, then add or subtract exactly as you would with whole numbers. Bring the decimal point straight down into the answer. A whole number has an invisible decimal point at its right end, so 20 can be written 20.00 when you subtract from it.
Worked Example: Find 4.6 + 12.35 + 0.078, and then 20 − 7.46.
Line up the points and pad with zeros (4.600, 12.350, 0.078):
4.600 12.350 + 0.078 ________ 17.028
The sum is 17.028. For the subtraction, write 20 as 20.00 and borrow across the zeros:
20.00 − 7.46 _______ 12.54
The difference is 12.54. Check by adding: 12.54 + 7.46 = 20.00.
1. Find 3.7 + 2.58.
Explanation: Write 3.70 + 2.58 = 6.28. The wrong choices come from not lining up the decimal points.
2. Find 15.2 − 8.74.
Explanation: Write 15.20 − 8.74 = 6.46. Check: 6.46 + 8.74 = 15.20.
3. Find 0.5 + 0.25 + 3.
Explanation: Write 0.50 + 0.25 + 3.00 = 3.75.
4. Maria buys two items that cost $12.49 and $8.75 and pays with a $30 bill. How much change should she get?
Explanation: The items cost 12.49 + 8.75 = $21.24. The change is 30.00 − 21.24 = $8.76. ($21.24 is the total, not the change.)
5. A rain gauge holds 1.125 inches at 1 p.m. and 1.875 inches at 3 p.m. How much rain fell between 1 p.m. and 3 p.m.?
Explanation: Subtract: 1.875 − 1.125 = 0.750 inch.
To multiply decimals, ignore the decimal points and multiply as whole numbers. Then count the total number of decimal places in the two factors and place the decimal point so the answer has that many places. You do not line up the decimal points for multiplication. To divide decimals, make the divisor (the number you divide by) a whole number by moving its decimal point to the right. Move the decimal point in the dividend (the number being divided) the same number of places, then divide and place the point directly above its new position in the quotient. Use a zero as a placeholder if you need one.
Worked Example: Find 3.6 × 0.25 and 4.5 ÷ 0.15.
Multiply 36 × 25 = 900. The factors have 1 + 2 = 3 decimal places, so the answer is 0.900, which is 0.9.
For 4.5 ÷ 0.15, move both decimal points two places to the right: 450 ÷ 15 = 30. Check: 30 × 0.15 = 4.5.
1. Find 0.6 × 0.4.
Explanation: 6 × 4 = 24, and there are 2 decimal places in total, so 0.24.
2. Find 2.5 × 1.2.
Explanation: 25 × 12 = 300 with 2 decimal places gives 3.00, which is 3.
3. Find 8.4 ÷ 0.2.
Explanation: Move both points one place: 84 ÷ 2 = 42.
4. Jelly beans cost $9.60 per pound. How much do 1.5 pounds cost?
Explanation: 9.60 × 1.5 = 14.40, so $14.40.
5. A box of cereal costs $4.69 and holds 19 servings. To the nearest cent, what is the cost of one serving?
Explanation: 4.69 ÷ 19 ≈ 0.2468. The thousandths digit 6 rounds the hundredths up, so $0.25.
A fraction names equal parts of a whole: the denominator tells how many equal parts the whole is split into, and the numerator tells how many of them you have. A proper fraction is less than 1 (numerator smaller), an improper fraction is 1 or more (numerator equal to or larger), and a mixed number is a whole number plus a proper fraction. To change an improper fraction to a mixed number, divide the numerator by the denominator: the quotient is the whole number and the remainder goes over the denominator. To go the other way, multiply the whole number by the denominator, add the numerator, and keep the denominator. Equivalent fractions have the same value; you make them by multiplying or dividing the numerator and denominator by the same number. A fraction is in lowest terms when the only common factor is 1. To compare fractions, rewrite them with a common denominator and compare the numerators.
Worked Example: (a) Write 23/5 as a mixed number and 3 2/7 as an improper fraction. (b) Reduce 18/24. (c) Which is greater, 5/8 or 7/12?
(a) 23 ÷ 5 = 4 remainder 3, so 23/5 = 4 3/5. And 3 2/7 = (3 × 7 + 2)/7 = 23/7.
(b) The greatest common factor of 18 and 24 is 6, so 18/24 = 3/4.
(c) A common denominator is 24: 5/8 = 15/24 and 7/12 = 14/24. Since 15 > 14, 5/8 is greater.
1. Write 29/6 as a mixed number.
Explanation: 29 ÷ 6 = 4 remainder 5, so 29/6 = 4 5/6.
2. Write 5 3/8 as an improper fraction.
Explanation: (5 × 8) + 3 = 43, so the fraction is 43/8.
3. Reduce 42/56 to lowest terms.
Explanation: The greatest common factor is 14: 42 ÷ 14 = 3 and 56 ÷ 14 = 4. So 3/4. (21/28 and 6/8 are equivalent but not in lowest terms.)
4. Which fraction has the greatest value?
Explanation: As decimals: 2/3 ≈ 0.667, 3/5 = 0.6, 5/8 = 0.625, 7/12 ≈ 0.583. The greatest is 2/3.
5. What number makes 3/5 = ?/35 true?
Explanation: 5 × 7 = 35, so multiply the numerator by 7 too: 3 × 7 = 21.
You can only add or subtract fractions that have the same denominator. With like denominators, add or subtract the numerators and keep the denominator. With unlike denominators, first find a least common denominator (LCD), the smallest number both denominators divide into, and rewrite each fraction with it. For mixed numbers, work with the fraction parts first, then the whole numbers. If the fraction part of the answer is improper, carry the extra whole number over. When you subtract and the first fraction is smaller than the second, borrow 1 from the whole number and add it to the fraction as a fraction equal to 1 (such as 12/12). Always reduce the answer and write an improper fraction as a mixed number.
Worked Example: Find 3/4 + 5/6 and then 7 1/3 − 2 3/4.
The LCD of 4 and 6 is 12. 3/4 = 9/12 and 5/6 = 10/12, so the sum is 19/12 = 1 7/12.
For the subtraction, the LCD of 3 and 4 is 12, so 7 1/3 = 7 4/12 and 2 3/4 = 2 9/12. Since 4/12 is less than 9/12, borrow: 7 4/12 = 6 16/12. Now 6 16/12 − 2 9/12 = 4 7/12, so the answer is 4 7/12. Check: 4.583… + 2.75 = 7.333…
1. Find 2/5 + 3/10.
Explanation: 2/5 = 4/10, so 4/10 + 3/10 = 7/10.
2. Find 5/6 − 1/4.
Explanation: The LCD is 12: 10/12 − 3/12 = 7/12.
3. Find 3 1/2 + 2 3/4.
Explanation: 3 2/4 + 2 3/4 = 5 5/4 = 6 1/4.
4. Find 9 − 4 3/8.
Explanation: Borrow 1 from 9: 8 8/8 − 4 3/8 = 4 5/8.
5. Craig glues a piece of oak that is 5/16 inch thick to a piece of pine that is 7/8 inch thick. What is the total thickness?
Explanation: 7/8 = 14/16, so 5/16 + 14/16 = 19/16 = 1 3/16 inches.
6. A bolt is 2 7/8 inches long and another bolt is 1 1/4 inches long. How much longer is the first bolt?
Explanation: 1 1/4 = 1 2/8. Then 2 7/8 − 1 2/8 = 1 5/8 inches.
You do not need a common denominator to multiply fractions. Multiply the numerators, multiply the denominators, and reduce. A shortcut called canceling lets you divide a numerator and a denominator (from any of the fractions) by the same number before you multiply. Write whole numbers as fractions over 1, and change mixed numbers to improper fractions first. To divide fractions, invert the divisor (the fraction you divide by) so that its numerator and denominator trade places, change the division sign to multiplication, and multiply. Multiplying by a fraction less than 1 makes the number smaller because you are taking a “part of” it, while dividing by a fraction less than 1 makes the number larger because you are counting how many small pieces fit.
Worked Example: Find 1 1/2 × 1 1/5 and then 3 3/4 ÷ 1 2/3.
Change to improper fractions: 3/2 × 6/5. Cancel the common factor 2 (3/2 and 6/5 become 3/1 and 3/5) to get 3 × 3/(1 × 5) = 9/5 = 1 4/5.
For the division: 15/4 ÷ 5/3 = 15/4 × 3/5. Cancel 5 into 15 to get 3/4 × 3/1 = 9/4 = 2 1/4.
Quick sense check: 8 ÷ 1/2 = 8 × 2 = 16, because eight wholes hold sixteen halves.
1. Find 2/3 × 3/8.
Explanation: Multiply across: 6/24 = 1/4. (Or cancel first: the 3s cancel and 2/8 = 1/4.)
2. Find 4 1/2 × 2/3.
Explanation: 9/2 × 2/3 = 18/6 = 3.
3. Find 5/6 ÷ 2/3.
Explanation: 5/6 × 3/2 = 15/12 = 5/4 = 1 1/4.
4. Find 12 ÷ 3/4.
Explanation: 12/1 × 4/3 = 48/3 = 16.
5. Find 6 2/3 ÷ 1 1/4.
Explanation: 20/3 × 4/5 = 80/15 = 16/3 = 5 1/3.
6. A tailor has 25 yards of fabric. Each shirt needs 3 1/4 yards. How many complete shirts can she make?
Explanation: 25 ÷ 13/4 = 25 × 4/13 = 100/13 ≈ 7.69. Only complete shirts count, so 7.
Fractions and decimals are two ways to write the same part of a whole. To change a fraction to a decimal, divide the numerator by the denominator. To change a decimal to a fraction, write the digits without the decimal point over the place value of the last digit, then reduce. Memorizing the common equivalents below saves time on the GED test, because you can often multiply by a simple fraction instead of a messy decimal, and they also help with percents. Decimals such as 0.333… and 0.666… repeat forever and are written as the fractions 1/3 and 2/3.
| Decimal | Fraction | Decimal | Fraction | Decimal | Fraction |
|---|---|---|---|---|---|
| 0.1 | 110 | 0.4 | 25 | 0.75 | 34 |
| 0.125 | 18 | 0.5 | 12 | 0.8 | 45 |
| 0.2 | 15 | 0.6 | 35 | 0.875 | 78 |
| 0.25 | 14 | 0.625 | 58 | 0.9 | 910 |
| 0.3 | 310 | 0.7 | 710 | 0.333… | 13 |
| 0.375 | 38 | 0.666… | 23 |
Worked Example: (a) Write 3/8 as a decimal and 0.35 as a fraction. (b) A cough syrup dose is 0.25 ounce. How many ounces are in 48 doses? (c) A worker has 40 hours to inspect 12 machines. How many hours per machine?
(a) 3 ÷ 8 = 0.375. The last digit of 0.35 is in the hundredths place, so 0.35 = 35/100 = 7/20.
(b) Since 0.25 = 1/4, find 1/4 of 48: 48 ÷ 4 = 12 ounces.
(c) 40 ÷ 12 shows 3.3333… on a calculator. Because 0.333… = 1/3, the answer is 3 1/3 hours.
1. Write 5/8 as a decimal.
Explanation: 5 ÷ 8 = 0.625.
2. Write 0.45 as a fraction in lowest terms.
Explanation: 0.45 = 45/100. Dividing top and bottom by 5 gives 9/20.
3. Write 0.375 as a fraction in lowest terms.
Explanation: 0.375 = 375/1000, and dividing by 125 gives 3/8 (it is also in the table).
4. A calculator shows 5.3333333 hours for a job. Which mixed number is this?
Explanation: The repeating 0.333… equals 1/3, so the time is 5 1/3 hours.
5. A sale takes 0.125 off the price of an item. What is the discount on an item priced at $64?
Explanation: 0.125 = 1/8, and 64 ÷ 8 = $8.
A number line shows numbers in order. Numbers to the right are greater and numbers to the left are smaller, and arrows on the ends show that the line goes on forever. To read a point, first find the two labeled marks on either side, count how many equal steps lie between them, and divide the distance by that number to get the value of one step. The same idea works for decimals, fractions, and mixed numbers, and some questions ask you to place a dot on the line instead. If the labels are fractions, it often helps to turn the choices into decimals, or all into fractions with the same denominator, so you can compare them.
Worked Example: On a number line from 0 to 2 the tick marks are every 1/4. Where is the point 0.75?
Since 0.75 = 3/4, the point is the third tick mark to the right of 0. It is greater than 1/2 and less than 1.
1. What is the value of the point on the number line below?
Explanation: Between 2 and 3 there are 5 equal steps of 0.2 each. The labeled marks are 2, 2.2, 2.4, 2.6 and 3, so the unlabeled mark just before 3 is 2.8.
2. What is the value of the point on the number line below?
Explanation: The labeled marks are in steps of 1/3. The point is one step after 1/3, so it is at 2/3.
3. Which decimal lies between 1/4 and 1/2 on a number line?
Explanation: 1/4 = 0.25 and 1/2 = 0.5. Only 0.3 is between them; 0.5 is exactly 1/2.
4. Which fraction would be farthest to the right on a number line?
Explanation: As decimals: 0.75, 0.625, 0.7, 0.667. The greatest is 3/4, so it is the farthest right.
5. Angela baked 24 cookies and gave away 16. Which fraction of the cookies, in lowest terms, would be marked on a number line?
Explanation: 16/24 reduces by 8 to 2/3.
The GED test gives you an on-screen version of the TI-30XS MultiView calculator for most of the Mathematical Reasoning Test. Decimal operations are typed in the same order as whole-number operations; use the decimal point key (.) under the 2 for the decimal point. For fractions, use the fraction key (n/d) and move between the numerator and denominator with the cursor keys: the down arrow goes to the denominator and the right arrow leaves the fraction. For mixed numbers, press the green 2nd key and then the fraction key (U n/d) before entering the whole number, numerator, and denominator. The toggle key switches an answer between its fraction and decimal forms, and the same 2nd function key changes an improper fraction into a mixed number.
| To solve | Press | Display |
|---|---|---|
| 6.45 + 3.8 | 6.45 [+] 3.8 [enter] | 10.25 |
| 7.2 − 0.65 | 7.2 [−] 0.65 [enter] | 6.55 |
| 0.8 × 15 | 0.8 [×] 15 [enter] | 12 |
| 2.4 ÷ 4 | 2.4 [÷] 4 [enter] | 0.6 |
Worked Example: (a) Reduce 56/448 and then write the answer as a decimal. (b) A plastic pipe 20 5/8 feet long is cut into pieces that are 1 7/8 feet long. How many pieces are there?
(a) Clear the calculator, press the fraction key, type 56, press the down arrow, type 448, and press enter. The calculator reduces it to 1/8. Press the toggle key to see 0.125.
(b) Enter 20 5/8 ÷ 1 7/8 using the mixed number function for each number (type the whole number, move right, type the numerator, move down, type the denominator, move right). The answer shows 11 pieces. You can check by hand: 165/8 ÷ 15/8 = 165/15 = 11.
1. Use a calculator to find 4.2 + 1.87 + 2.009.
Explanation: 4.200 + 1.870 + 2.009 = 8.079.
2. Find 5 1/3 − 2 5/12.
Explanation: 5 4/12 − 2 5/12 = 4 16/12 − 2 5/12 = 2 11/12.
3. What is $18.75 × 12?
Explanation: 18.75 × 12 = 225.00.
4. Find 9 3/8 ÷ 1 1/4.
Explanation: 75/8 × 4/5 = 300/40 = 7.5 = 7 1/2. Choosing 11 23/32 means multiplying by 5/4 instead of inverting to 4/5.
5. The yearly insurance on a house is found by multiplying the sale price by 0.0125. What is the cost for a house priced at $118,000?
Explanation: 118,000 × 0.0125 = 1,475, so $1,475.
GED word problems often combine several skills in one question. A reliable approach is to (1) read the question and underline what you are asked to find, (2) pick out the numbers you need, (3) decide the operation (adding or subtracting to combine or compare, multiplying for “of” or “each,” and dividing to split into equal groups or to see how many fit), (4) check whether the answer choices are decimals or fractions so you can choose the more convenient form, and (5) check that your answer is sensible. When a problem asks how many complete or whole items, round down, and for an average, add the values and divide by how many there are.
Worked Example: A wildlife preserve makes up 3/8 of a 64-acre park. How many acres is the preserve?
“Of” means multiply: 3/8 × 64 = 3 × 8 = 24 acres. The answer is less than 64, which makes sense.
1. A wooden strip is 18 1/4 inches long. After 3 1/2 inches are cut from one end, what is the new length?
Explanation: 18 1/4 = 17 5/4, and 17 5/4 − 3 2/4 = 14 3/4 inches.
2. A computer is sold with $150 down and 12 monthly payments of $89.50. What is the total cost?
Explanation: 12 × 89.50 = 1,074.00. Add the down payment: 1,074 + 150 = $1,224.00.
3. Three packages weigh 1 1/4 pounds, 3 3/4 pounds, and 2 1/2 pounds. What is the average weight?
Explanation: The total is 1.25 + 3.75 + 2.5 = 7.5 pounds. Divide by 3 packages: 2.5 pounds. (7.5 is the total, not the average.)
4. A clinic physical takes 1/3 hour. How many physicals can a worker complete in a 3 1/2-hour morning?
Explanation: 3 1/2 ÷ 1/3 = 7/2 × 3 = 21/2 = 10.5. Only complete physicals count, so 10.
5. A stadium has 6000 seats and 4500 are sold. What fraction of the seats were sold?
Explanation: 4500/6000 = 45/60 = 3/4.
6. Gina earns $9 an hour and earned $258 in one week. How many hours did she work?
Explanation: 258 ÷ 9 = 28.666…, and 0.666… = 2/3, so she worked 28 2/3 hours.
Divide the numerator by the denominator. For example, 3/8 means 3 ÷ 8, which is 0.375. Some fractions, like 1/3, give a repeating decimal (0.333…).
Write the digits without the decimal point over the place value of the last digit, then reduce. For example, 0.35 is 35/100, which reduces to 7/20.
Only when you add or subtract fractions. For multiplication and division you never need one, but you may cancel common factors to keep the numbers small.
Dividing by a number is the same as multiplying by its reciprocal. For example, 8 ÷ 1/2 asks how many halves are in 8, which is 8 × 2 = 16.
Look at the answer choices. If they are fractions or mixed numbers, give the answer in that form; if they are decimals, use decimals. Memorizing common equivalents makes it easy to switch.
Yes. The Mathematical Reasoning Test provides an on-screen TI-30XS MultiView calculator for most questions, and this lesson shows how to use the fraction and toggle keys. Still, learn to do the work by hand so you can check results.
The GED Math practice in the GED Math QBank gives you more practice on this topic, and you can browse every other topic in the full GED Math QBank.
