Convert Number Forms - TEAS Mathematics

1. A recipe requires $\frac{5}{8}$ cup of flour. A baker has a container with 3.5 cups of flour. After making the recipe, the baker uses 20% of the remaining flour for dusting. What fraction of the original 3.5 cups of flour was used for dusting?
   Answer: $\frac{23}{140}$ First, convert all values to a common format, like decimals. The recipe uses $\frac{5}{8} = 0.625$ cups. The original amount is 3.5 cups. The remaining flour is $3.5 - 0.625 = 2.875$ cups. The baker uses 20% of the remaining flour for dusting, which is $0.20 \times 2.875 = 0.575$ cups. The question asks for this amount as a fraction of the original 3.5 cups. The fraction is $\frac{0.575}{3.5}$. To simplify, multiply the numerator and denominator by 1000 to remove decimals: $\frac{575}{3500}$. Simplify this fraction by dividing the numerator and denominator by their greatest common divisor, 25: $\frac{575 \div 25}{3500 \div 25} = \frac{23}{140}$.
2. A student's scores on four assignments were 85.1%, $\frac{13}{15}$, 0.86, and $\frac{6}{7}$. Which of the following lists the scores from greatest to least?
   Answer: $\frac{13}{15}, 0.86, \frac{6}{7}, 85.1\%$ To compare the scores, convert all of them to decimals with at least three places for accuracy. $85.1\% = 0.851$. $\frac{13}{15} = 13 \div 15 = 0.8666... = 0.8\overline{6}$. $0.86 = 0.860$. $\frac{6}{7} = 6 \div 7 \approx 0.857$. Now, compare the decimal values: 0.866..., 0.860, 0.857, 0.851. The order from greatest to least is $\frac{13}{15}$, then 0.86, then $\frac{6}{7}$, and finally 85.1%.
3. A solution contains $\frac{3}{11}$ grams of salt per liter. A procedure requires using 1.5 liters of this solution, but the concentration must first be increased by 27.5%. What is the total mass of salt, in grams, in the final 1.5-liter solution, expressed as a decimal rounded to the nearest hundredth?
   Answer: 0.52 First, find the initial mass of salt in 1.5 liters. Mass = $\frac{3}{11} \text{ g/L} \times 1.5 \text{ L} = \frac{4.5}{11}$ grams. The concentration is then increased by 27.5%. This means the new mass will be $100\% + 27.5\% = 127.5\%$ of the initial mass. Convert 127.5% to a decimal: 1.275. The new mass is $1.275 \times \frac{4.5}{11}$. It's easier to compute the fraction first: $\frac{4.5}{11} \approx 0.40909...$ grams. Now, multiply by 1.275: $1.275 \times 0.40909... \approx 0.52159...$ grams. Rounding to the nearest hundredth gives 0.52 grams.
4. An investment of $540 decreases in value by $\frac{1}{9}$. The new value then increases by \(11.\overline{1}\%. What is the final value of the investment?
   Answer: $533.33 First, calculate the decrease. The decrease is $\frac{1}{9} \times 540 = \$60$. The value after the decrease is $540 - 60 = \$480$. Next, the new value increases by 11.\overline{1}\%. It is crucial to recognize that the repeating decimal \(0.111... is the fraction $\frac{1}{9}$. Therefore, $11.\overline{1}\%$ is equivalent to $\frac{1}{9}$ as a rate. The increase is calculated on the new value of $480: $\frac{1}{9} \times 480 = \frac{480}{9} = 53.\overline{3}$, or $53.33. The final value is $\$480 + \$53.33 = \$533.33$.
5. In a survey, $\frac{3}{8}$ of participants preferred Brand A. Of the remaining participants, 65% preferred Brand B, and the rest preferred Brand C. What decimal represents the proportion of all participants who preferred Brand C?
   Answer: 0.21875 First, find the fraction of participants who did not prefer Brand A. This is $1 - \frac{3}{8} = \frac{5}{8}$. Of this remaining group, 65% preferred Brand B, which means $100\% - 65\% = 35\%$ preferred Brand C. To find the proportion of the total who preferred Brand C, we must calculate 35% of $\frac{5}{8}$. Convert both to decimals: $35\% = 0.35$ and $\frac{5}{8} = 0.625$. Now multiply: $0.35 \times 0.625 = 0.21875$.
6. A patient's medication dosage is reduced by 0.45%. Which fraction represents this reduction in its simplest form?
   Answer: $\frac{9}{2000}$ To convert a percentage to a fraction, first convert it to a decimal by dividing by 100. $0.45\% = 0.45 \div 100 = 0.0045$. Next, convert the decimal to a fraction. The decimal 0.0045 is read as 'forty-five ten-thousandths', which is written as $\frac{45}{10000}$. Finally, simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is 5. $\frac{45 \div 5}{10000 \div 5} = \frac{9}{2000}$.
7. A company allocates $\frac{1}{8}$ of its budget to marketing. Within the marketing budget, 32% is allocated for digital ads. The CEO decides to cut the digital ad budget by 12.5% of its current value. What fraction of the total company budget is the new, reduced digital ad budget?
   Answer: $\frac{7}{200}$ First, find the initial digital ad budget as a fraction of the total budget. This is 32% of $\frac{1}{8}$. Convert 32% to a fraction: $\frac{32}{100} = \frac{8}{25}$. So, the digital ad budget is $\frac{8}{25} \times \frac{1}{8} = \frac{1}{25}$ of the total budget. Next, this budget is cut by 12.5%. A 12.5% cut means the remaining amount is $100\% - 12.5\% = 87.5\%$ of the original digital ad budget. Convert 87.5% to a fraction: $0.875 = \frac{875}{1000} = \frac{7}{8}$. The new digital ad budget is $\frac{7}{8}$ of the old one. So, the new budget as a fraction of the total is $\frac{7}{8} \times \frac{1}{25} = \frac{7}{200}$.
8. A machine part expands by a length of 0.03 meters and contracts by a length of $0.0\overline{27}$ meters due to temperature changes. What is the net change in length, expressed as a simplified fraction?
   Answer: $\frac{3}{1100}$ The net change is the difference between the expansion and contraction: $0.03 - 0.0\overline{27}$. To subtract accurately, convert both decimals to fractions. $0.03 = \frac{3}{100}$. To convert the repeating decimal $0.0\overline{27}$, let $x = 0.0\overline{27}$. Then $10x = 0.\overline{27}$ and $1000x = 27.\overline{27}$. Subtracting these gives $990x = 27$, so $x = \frac{27}{990}$, which simplifies to $\frac{3}{110}$. Now subtract the fractions: $\frac{3}{100} - \frac{3}{110}$. The least common denominator is 1100. $\frac{33}{1100} - \frac{30}{1100} = \frac{3}{1100}$.
9. What is $1\frac{1}{6}\%$ of 240, expressed as a decimal?
   Answer: 2.8 First, convert the mixed number percentage to an improper fraction: $1\frac{1}{6}\% = \frac{7}{6}\%$. To use this in a calculation, convert the percentage to a standard fraction or decimal by dividing by 100: $\frac{7}{6} \div 100 = \frac{7}{600}$. Now, find $\frac{7}{600}$ of 240. This is $\frac{7}{600} \times 240 = \frac{7 \times 240}{600} = \frac{7 \times 24}{60}$. Simplifying the fraction $\frac{24}{60}$ gives $\frac{2}{5}$. The calculation is now $7 \times \frac{2}{5} = \frac{14}{5}$. Finally, convert this fraction to a decimal: $14 \div 5 = 2.8$.
10. In City A, the property tax is $\frac{3}{160}$ of a property's assessed value. For a property assessed at $320,000 in City A, the tax is the same as the tax on a property assessed at $240,000 in City B. What is the property tax rate in City B, expressed as a percentage?
    Answer: 2.5% First, calculate the property tax in City A. Tax = $\frac{3}{160} \times 320,000$. This simplifies to $3 \times \frac{320,000}{160} = 3 \times 2000 = \$6,000$. This $6,000 is also the tax on the property in City B. To find the tax rate in City B, divide the tax by the assessed value: Rate = $\frac{6,000}{240,000}$. This fraction simplifies to $\frac{6}{240} = \frac{1}{40}$. To express this rate as a percentage, convert the fraction to a decimal ($1 \div 40 = 0.025$) and then multiply by 100. $0.025 \times 100 = 2.5\%$.
11. A container holds 72 liters of a chemical. A scientist removes $\frac{2}{9}$ of the chemical. A second scientist adds back an amount equal to 22.5% of the original amount. What is the final volume in the container, expressed as a decimal?
    Answer: 72.2 First, calculate the amount removed: $\frac{2}{9} \times 72 = 2 \times 8 = 16$ liters. The volume remaining is $72 - 16 = 56$ liters. Next, calculate the amount added back. This is 22.5% of the original amount (72 L). Convert 22.5% to a decimal: 0.225. The amount added is $0.225 \times 72 = 16.2$ liters. The final volume is the remaining volume plus the added volume: $56 + 16.2 = 72.2$ liters.
12. Express 37% as a decimal.
    Answer: 0.37 Converting percentages to decimals is a fundamental skill that appears frequently on the TEAS exam. When you see a percentage conversion problem, remember that "percent" literally means "per hundred," so you're working with parts of 100. To convert 37% to a decimal, you divide by 100, which shifts the decimal point two places to the left. Starting with 37%, you can think of this as 37.0%. Moving the decimal point two places left gives you 0.37. Alternatively, you can set up the conversion as: $37\% = \frac{37}{100} = 0.37$ Looking at the wrong answers reveals common conversion mistakes. Choice A (0.037) results from moving the decimal point three places to the left instead of two—this would be the answer if you were converting 3.7% to a decimal. Choice C (3.7) comes from moving the decimal point one place to the left, which is the error students make when they forget the "divide by 100" rule and instead divide by 10. Choice D (37.0) represents no conversion at all—simply writing the percentage as a whole number without any decimal adjustment. The correct answer is B (0.37). For percentage-to-decimal conversions, always remember the "two-place rule": move the decimal point exactly two places to the left and drop the percent sign. Practice this with common percentages like 50% = 0.50, 25% = 0.25, and 100% = 1.00 to build your confidence with the pattern.
13. A quantity of a substance is $0.58\overline{3}$ of its original mass of 120 kg. If 75% of this quantity is removed, what is the remaining mass in kg?
    Answer: 17.5 kg First, convert the repeating decimal $0.58\overline{3}$ to a fraction. Let $x = 0.58\overline{3}$. Then $100x = 58.\overline{3}$ and $1000x = 583.\overline{3}$. Subtracting gives $900x = 525$, so $x = \frac{525}{900}$, which simplifies to $\frac{7}{12}$. The quantity of the substance is $\frac{7}{12} \times 120$ kg = $7 \times 10 = 70$ kg. If 75% of this quantity is removed, then $100\% - 75\% = 25\%$ remains. The remaining mass is 25% of 70 kg. $0.25 \times 70 = 17.5$ kg.
14. Which of the following two quantities is greater, and by what percentage of the smaller quantity? Quantity X: $37.5\%$ of $\frac{4}{5}$ of 200. Quantity Y: $\frac{7}{8}$ of $32\%$ of 100.
    Answer: Quantity X is greater by approximately 114.3% First, calculate both quantities. For Quantity X: $37.5\% = \frac{3}{8}$. $\frac{4}{5}$ of 200 is $\frac{4}{5} \times 200 = 160$. So, X = $\frac{3}{8} \times 160 = 3 \times 20 = 60$. For Quantity Y: $32\%$ of 100 is 32. So, Y = $\frac{7}{8} \times 32 = 7 \times 4 = 28$. Comparing them, Quantity X (60) is greater than Quantity Y (28). The difference is $60 - 28 = 32$. The question asks for this difference as a percentage of the smaller quantity (Y). The percentage is $\frac{\text{difference}}{\text{smaller value}} \times 100 = \frac{32}{28} \times 100$. $\frac{32}{28} = \frac{8}{7} \approx 1.1428$. As a percentage, this is approximately 114.3%.
15. After a 15% discount, the price of an item is $42.50. Before the discount, the price was $\frac{5}{4}$ of the wholesale cost. What is the wholesale cost as a percentage of the final discounted price? Round to the nearest whole percent.
    Answer: 94% This is a multi-step problem. First, find the original price before the discount. A 15% discount means the final price is $100\% - 15\% = 85\%$ of the original price. So, $0.85 \times \text{Original Price} = 42.50$. The original price is $42.50 \div 0.85 = \$50$. Next, find the wholesale cost (W). The original price ($50) was $\frac{5}{4}$ of W: $50 = \frac{5}{4}W$. Solving for W gives $W = 50 \times \frac{4}{5} = \$40$. Finally, express the wholesale cost ($40) as a percentage of the final discounted price ($42.50): \frac{40}{42.50} \times 100\%. \(40 \div 42.50 \approx 0.94117.... As a percentage, this is 94.117...%, which rounds to 94%.
16. Which of the following is $0.875$ written as a fraction in simplest form?
    Answer: $\dfrac{7}{8}$ When converting decimals to fractions, you need to understand place value and how to simplify fractions to lowest terms. To convert $0.875$ to a fraction, recognize that this decimal has three places after the decimal point, so it represents $\frac{875}{1000}$. However, the question asks for simplest form, which means you must reduce this fraction by finding the greatest common factor (GCF) of 875 and 1000. Finding the GCF: Both 875 and 1000 are divisible by 125. When you divide: $875 ÷ 125 = 7$ and $1000 ÷ 125 = 8$. This gives you $\frac{7}{8}$. Since 7 and 8 share no common factors other than 1, this fraction is in simplest form. You can verify: $7 ÷ 8 = 0.875$ ✓ Now examining each choice: Choice A ($\frac{7}{8}$) is correct as shown above. Choice B ($\frac{9}{10} = 0.9$) represents a different decimal value entirely. Choice C ($\frac{875}{1000}$) does equal $0.875$, but it's not in simplest form since both numerator and denominator can be divided by 125. Choice D ($\frac{13}{16} = 0.8125$) also represents a different decimal value. On the TEAS, decimal-to-fraction problems often include the unreduced fraction as a distractor. Always check if your answer can be simplified further by finding common factors between the numerator and denominator. Practice recognizing common decimal-fraction equivalents like $0.875 = \frac{7}{8}$ to save time.

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