Math 1 Quiz: Estimating For Reasonableness

What this quiz covers

This quiz focuses on Estimating For Reasonableness, giving you a quick way to practice the rules, question types, and explanations that matter most for Math 1.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. The decimal result suggests an error since pills cannot be divided into fractional parts in practice
2. Estimating: $180 \div 7 \approx 180 \div 6 = 30$, so 25.71 is too low by about 4-5 pills per patient (correct answer)
3. The calculation gives a remainder, which means the division wasn't performed correctly for this context
4. Checking: $25.71 \times 7 = 179.97 \approx 180$, so the mathematical calculation appears correct

Question 2

1. Round both dimensions to the nearest ten and multiply: $50 \times 20 = 1,000$ square feet (correct answer)
2. Round both dimensions to the nearest whole number and multiply: $48 \times 23 = 1,104$ square feet
3. Round the first dimension up and the second down: $50 \times 20 = 1,000$ square feet
4. Use the original dimensions but round the final answer: $1,109$ square feet

Question 3

1. The calculation is reasonable since 149.7 falls within the manufacturer's range of 140-160 square feet per gallon (correct answer)
2. Recalculation is needed because the coverage rate is at the low end of the expected range
3. The answer needs verification since quick estimation gives 2,400 ÷ 16 = 150, which is very close to 149.7
4. The calculation appears incorrect since 2,350 ÷ 15 ≈ 157, which differs from 149.7

Question 4

1. Since 20 is less than twice 12, the flour needed should be less than $2 \times 2\frac{3}{4} = 5\frac{1}{2}$ cups
2. Round the original amount to 3 cups, then calculate $\frac{20}{12} \times 3 = 5$ cups as the estimate
3. Since $\frac{20}{12} = 1\frac{2}{3}$, multiply: $1\frac{2}{3} \times 2\frac{3}{4} \approx \frac{5}{3} \times \frac{11}{4} \approx 4\frac{1}{2}$ cups
4. Convert everything to decimals: $2.75 \times \frac{20}{12} = 2.75 \times 1.67 \approx 4.6$ cups (correct answer)

Question 5

1. The calculation is mathematically correct, and the difference is due to route variations and traffic conditions
2. Estimation using $70 \times 4.5 = 315$ miles confirms the calculation is reasonable, but GPS measures actual path (correct answer)
3. The calculation assumes constant speed, while GPS measures actual distance including stops and speed changes
4. Both answers are reasonable given measurement precision, with about 4% difference between calculated and actual distance

Question 6

1. Convert to decimals: $\frac{5}{7} \approx 0.71$ and $\frac{3}{11} \approx 0.27$, so $0.71 - 0.27 = 0.44$, which matches closely (correct answer)
2. Use benchmark fractions: $\frac{5}{7} \approx \frac{1}{2}$ and $\frac{3}{11} \approx \frac{1}{4}$, so $\frac{1}{2} - \frac{1}{4} = \frac{1}{4} = 0.25$, indicating an error
3. Compare to 1: since both fractions are less than 1, their difference must be less than 1, so 0.4455 is reasonable
4. Cross multiply: $5 \times 11 = 55$ and $3 \times 7 = 21$, so $55 - 21 = 34$, confirming the result

Question 7

1. Since $\sqrt{196} = 14$ and $197 > 196$, the answer should be slightly larger than 14, so 14.04 is reasonable
2. Since $\sqrt{200} = 10\sqrt{2} \approx 14.14$ and $197 < 200$, the answer should be slightly smaller, so 14.04 is reasonable
3. Since $14^2 = 196$ and $15^2 = 225$, and $197$ is much closer to $196$, the answer should be much closer to $14$, so $14.04$ is reasonable (correct answer)
4. Since $\sqrt{100} = 10$ and $\sqrt{400} = 20$, and $197$ is between these values, any answer between $10$ and $20$ is reasonable

Question 8

1. Round both dimensions to the nearest ten and multiply: $50 \times 20 = 1,000$ square feet
2. Round both dimensions to the nearest whole number and multiply: $48 \times 23 = 1,104$ square feet (correct answer)
3. Add the dimensions and double the result: $2(47.8 + 23.2) = 142$ square feet
4. Use the formula $\frac{47.8 + 23.2}{2} \times 2 = 71$ square feet

Question 9

1. Estimate yearly: Year 1: $1254$, Year 2: $1310$, Year 3: $1369$, which closely matches $1372.84$
2. Round the rate to 5%: $1200 \times (1.05)^3 \approx 1200 \times 1.16 = 1392$, which is reasonably close to $1372.84$
3. Use the rule of 72: Money doubles in $\frac{72}{4.5} = 16$ years, so in 3 years expect about $\frac{3}{16}$ growth, giving unreasonable result
4. Use simple interest: $1200 \times 0.045 \times 3 = 162$, so total $= 1200 + 162 = 1362$, which is close to $1372.84$ (correct answer)

Question 10

1. Compare to the distance from origin: point $A$ is $\sqrt{58} \approx 7.6$ from origin, so $11.31$ seems reasonable for $AB$
2. Use the sum of horizontal and vertical distances: $|5-(-3)| + |(-1)-7| = 8 + 8 = 16$, so $11.31$ is too small
3. Check if the distance is longer than each coordinate difference: since $8 < 11.31$ and $8 < 11.31$, the answer is reasonable
4. Estimate using a right triangle: horizontal distance $\approx 8$, vertical distance $\approx 8$, so hypotenuse $\approx 8\sqrt{2} \approx 11.3$ (correct answer)

Question 11

1. Using $\pi \approx 3$: Volume $\approx 3 \times 2^2 \times 7 = 84$ cubic meters, so the answer is reasonable
2. Using $\pi \approx 3.14$: Volume $\approx 3.14 \times 2^2 \times 7 = 87.92$ cubic meters, so the answer is reasonable (correct answer)
3. Using surface area instead: $2\pi r^2 + 2\pi rh \approx 2(3)(4) + 2(3)(2)(7) = 108$, so recalculation is needed
4. Using $\pi \approx 3.14$: Volume $\approx 3.14 \times (2.1 + 6.8)^2 = 246.49$ cubic meters, so recalculation is needed

Question 12

1. Evaluate signs: positive times negative divided by positive should give a negative result, so $-4.5$ has the correct sign
2. Simplify step by step: $81 \times (-8) \div 36 = -648 \div 36 = -18$, which differs significantly from $-4.5$ (correct answer)
3. Use absolute values: $3^4 \times 2^3 \div 6^2 \approx 80 \times 8 \div 40 = 16$, then apply the negative sign
4. Round the bases: $(-3)^4 \approx 80$, $(-2)^3 \approx -8$, $(-6)^2 \approx 36$, giving $80 \times (-8) \div 36 \approx -18$

Question 13

1. Since $15^2 = 225$ and $20^2 = 400$, the answer should be between 15 and 20, so 17.2 is reasonable
2. Since $16^2 = 256$ and 247 is close to 256, the square root should be close to 16, making 17.2 too high (correct answer)
3. Since $15^2 = 225$ and $16^2 = 256$, and 247 is closer to 256, the answer should be closer to 16 than 15
4. Using the approximation $\sqrt{250} \approx 15.8$, the value 17.2 is significantly higher and needs recalculation

Question 14

1. Round both measurements to whole numbers: $\frac{1}{2} \times 9 \times 12 = 54$ square cm, which is about half the calculated answer
2. Use compatible numbers: $\frac{1}{2} \times 10 \times 12 = 60$ square cm, suggesting the answer is too high by nearly double
3. Round to one decimal place: $\frac{1}{2} \times 8.7 \times 12.3 \approx \frac{1}{2} \times 107 \approx 53.5$ square cm (correct answer)
4. Estimate using $9 \times 12 = 108$, then divide by 2 to get approximately $54$ square cm

Question 15

1. Simplify the powers of 10 first: $\frac{10^3}{10^2} = 10^1$, then calculate $\frac{5.6}{1.4} \times 10 \approx 4 \times 10 = 40$ (correct answer)
2. Convert to standard form: $5,600 \div 140 \approx 6,000 \div 100 = 60$, which contradicts the given answer
3. Use compatible numbers: $\frac{6 \times 10^3}{1.5 \times 10^2} = \frac{6}{1.5} \times 10^{3-2} = 4 \times 10^1 = 40$
4. Check by multiplication: $40 \times 10^1 \times 1.4 \times 10^2 = 56 \times 10^3$, which matches the original dividend

Question 16

1. Since all fractions are close to 1, the product should be close to $1 \times 1 \times 1 = 1$
2. Since each fraction is less than 1, the product should be much smaller than any individual fraction
3. Convert to decimals: $0.75 \times 0.89 \times 0.94$ should give approximately $0.6$ (correct answer)
4. Round each fraction to $\frac{1}{2}$: $\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8}$

Question 17

1. Since $2\frac{5}{6} \approx 3$ and $1\frac{1}{4} \approx 1$, the quotient should be close to $3 \div 1 = 3$
2. Convert to improper fractions and estimate: $\frac{17}{6} \div \frac{5}{4} \approx \frac{18}{6} \div \frac{6}{4} = 3 \div 1.5 = 2$ (correct answer)
3. Use the relationship: since $1\frac{1}{4} \times 2 = 2\frac{1}{2}$, and $2\frac{5}{6} > 2\frac{1}{2}$, the quotient should exceed 2
4. Round to mixed numbers with half denominators: $3 \div 1\frac{1}{2} = 2$, so $3\frac{1}{3}$ seems too high

Question 18

1. Convert all fractions to twenty-fourths: $\frac{14}{24} + \frac{15}{24} - \frac{4.5}{24} = \frac{24.5}{24}$, which doesn't match
2. Use decimal approximations: $0.58 + 0.63 - 0.19 = 1.02$, while $\frac{21}{24} \approx 0.88$ (correct answer)
3. Round to nearest half: $\frac{1}{2} + \frac{1}{2} - 0 = 1$, compared to $\frac{21}{24} \approx 0.9$
4. Since $\frac{7}{12} + \frac{5}{8} > 1$ and we subtract only $\frac{3}{16}$, the result should exceed 1

Question 19

1. $\frac{4 \times 8}{2} = 16$, so the answer appears reasonable and no recalculation is needed (correct answer)
2. $\frac{3 \times 8}{2} = 12$, so the answer appears unreasonable and recalculation is needed
3. $4 + 8 - 2 = 10$, so the answer appears unreasonable and recalculation is needed
4. $\frac{4 + 8}{2} = 6$, so the answer appears unreasonable and recalculation is needed

Question 20

1. The solutions are reasonable because they are close to 3 and -8, with small calculation errors likely due to rounding
2. The solutions need recalculation because $(x-3)(x+8) = 0$ gives exact solutions $x = 3$ and $x = -8$, not approximations (correct answer)
3. The solutions are reasonable because substituting them back gives values close to zero when accounting for rounding errors
4. The solutions need recalculation because the sum $3.2 + (-7.8) = -4.6$ doesn't equal the expected sum of $3 + (-8) = -5$