Arithmetic fluency

Recognize the structure. There are two different sign operations: subtracting −3 and multiplying 2 by −4.

Work it through. First multiply: 2(−4) = −8. Rewrite subtraction of a negative as addition:

−8 − (−3) + (−8) = −8 + 3 − 8 = −13.

Answer: −13.

Check. Starting at −8, moving right by 3 reaches −5, then moving left by 8 reaches −13.

Avoid the trap. A negative sign does not automatically turn positive when it appears near another negative sign. The rule depends on the operation: −8 + (−3) = −11, but −8 − (−3) = −5.

Key takeaway. Rewriting subtraction as addition of the opposite makes long signed expressions easier to audit.

Recognize the structure. The initial exponent applies to 4, not to the leading minus sign. Multiplication and division are handled left to right.

Work it through. The grouped difference is 4. Evaluate the powers, then multiply and divide:

−16 + 3(16) ÷ 8 = −16 + 48 ÷ 8 = −16 + 6 = −10.

Answer: −10.

Check. The second term is positive 6, so adding it to −16 gives a number between −16 and 0.

Avoid the trap. Changing −4² to (−4)² silently changes the problem. Also, do not divide 16 by 8 in a way that changes the grouping of an unrelated sum.

Key takeaway. Before calculating, identify the base of every power and the full numerator and denominator of every fraction.

Recognize the structure. Prime factorization reveals what the numbers share and what a common multiple must contain.

Work it through. Factor the numbers: 84 = 2² · 3 · 7 and 126 = 2 · 3² · 7. Use smaller exponents for the GCF and larger exponents for the LCM:

GCF = 2 · 3 · 7 = 42, LCM = 2² · 3² · 7 = 252.

Answer: GCF = 42; LCM = 252.

Check. 84/42 = 2 and 126/42 = 3. Also 252/84 = 3 and 252/126 = 2.

Avoid the trap. A common factor divides into each number; a common multiple is divisible by each number. Switching those directions changes the task.

Key takeaway. The GCF simplifies fractions; the LCM supplies an efficient common denominator. Here 84/126 = 2/3.

Recognize the structure. Twelfths and eighteenths are different-sized pieces. Convert them to the same size before adding.

Work it through. The least common denominator is 36. Multiply the first fraction by 3/3 and the second by 2/2: $\frac{\text{7}}{\text{12}}$ + $\frac{\text{5}}{\text{18}}$ = $\frac{\text{21}}{\text{36}}$ + $\frac{\text{10}}{\text{36}}$ = $\frac{\text{31}}{\text{36}}$.

Answer: 31/36.

Check. 7/12 ≈ 0.583 and 5/18 ≈ 0.278, so the sum is about 0.861. This matches 31/36 and is less than 1.

Avoid the trap. Adding the numerators and adding the denominators would give 12/30, which is smaller than 7/12 even though a positive amount was added. The magnitude check exposes the error immediately.

Key takeaway. A common denominator changes the number of pieces while preserving the value of each original fraction.

Recognize the structure. Converting both quantities to improper fractions avoids mixing whole-number subtraction with unlike fractional parts.

Work it through. Write 3$\frac{\text{1}}{\text{4}}$ = 13/4 and 1$\frac{\text{5}}{\text{6}}$ = 11/6. Use denominator 12:

$\frac{\text{13}}{\text{4}}$ − $\frac{\text{11}}{\text{6}}$ = $\frac{\text{39}}{\text{12}}$ − $\frac{\text{22}}{\text{12}}$ = $\frac{\text{17}}{\text{12}}$.

Answer: 17/12, also equal to 1$\frac{\text{5}}{\text{12}}$.

Check. 3.25 − 1.8333... = 1.4166..., consistent with 17/12. The result should lie between 1 and 2.

Avoid the trap. Subtracting 1 − 5 in the numerators while keeping a denominator of 4 or 6 does not subtract equal-sized parts.

Key takeaway. For an SPR answer, 17/12 fits the five-character positive limit; do not type the mixed-number form.

Recognize the structure. Everything in the numerator and denominator is multiplied, so common factors may be canceled across the product.

Work it through. Cancel a factor of 7 between 14 and 49 and a factor of 5 between 25 and 15:

$\frac{\text{14 · 25}}{\text{15 · 49}}$ = $\frac{\text{2 · 5}}{\text{3 · 7}}$ = $\frac{\text{10}}{\text{21}}$.

Answer: 10/21.

Check. Both original factors lie between 0 and 1, so the product should be smaller than either factor. 10/21 ≈ 0.476 meets that check.

Avoid the trap. Cancellation is division of the entire numerator and denominator by a common nonzero factor. It is not a rule for erasing matching digits.

Key takeaway. Factoring before multiplying prevents unnecessary large numbers and prepares you for rational algebraic expressions.

Recognize the structure. The divisor 5/8 is less than 1, so a positive quotient should be larger than 3/4.

Work it through. Multiply by the divisor’s reciprocal:

$\frac{\text{3}}{\text{4}}$ ÷ $\frac{\text{5}}{\text{8}}$ = $\frac{\text{3}}{\text{4}}$ · $\frac{\text{8}}{\text{5}}$ = $\frac{\text{6}}{\text{5}}$.

This asks how many 5/8-sized units make 3/4; the answer is 1.2 units.

Answer: 6/5.

Check. Multiplying back gives (6/5)(5/8) = 6/8 = 3/4.

Avoid the trap. Multiplying by 5/8 rather than 8/5 computes a product, not a quotient. “Flip a fraction” is incomplete advice: only the divisor is inverted.

Key takeaway. Always separate a mathematical quotient from a contextual full-item count. If only complete containers were allowed, a later rounding decision might be needed.

Recognize the structure. The large fraction bar groups the entire difference in its numerator. Calculate that difference first.

Work it through. The numerator is 8/12 − 3/12 = 5/12. Then divide by 5/6:

$\frac{\text{5}}{\text{12}}$ ÷ $\frac{\text{5}}{\text{6}}$ = $\frac{\text{5}}{\text{12}}$ · $\frac{\text{6}}{\text{5}}$ = $\frac{\text{1}}{\text{2}}$.

Answer: 1/2.

Check. (1/2)(5/6) = 5/12, which equals the original numerator.

Avoid the trap. An entry like 2/3-1/4/5/6 does not preserve the large fraction’s structure. A matching linear entry is (2/3-1/4)/(5/6).

Key takeaway. When an expression looks complicated, identify its outermost operation. Here it is division, with a complete group above and below the bar.

Recognize the structure. A quotient stays unchanged when its numerator and denominator are multiplied by the same nonzero number.

Work it through. Multiply both quantities by 1,000 to remove the decimals:

$\frac{\text{0.084}}{\text{0.35}}$ = $\frac{\text{84}}{\text{350}}$ = $\frac{\text{6}}{\text{25}}$ = 0.24.

Alternatively, multiplying both by 100 gives 8.4/35, the same quotient.

Answer: 0.24.

Check. 0.35(0.24) = 0.084. Since the divisor is smaller than 1, the quotient is larger than 0.084.

Avoid the trap. Moving the numerator’s decimal three places but the denominator’s only two changes the quotient by a factor of 10. Move both by the same factor.

Key takeaway. Equal scaling is the numerical form of multiplying a fraction by 1. It is also the reason unit-conversion factors can preserve a quantity’s value.

Recognize the structure. The three forms express the same part of the whole, but the percent symbol supplies a factor of 1/100.

Work it through. The fraction is 3/8. Dividing gives 0.375. To express that value as a percent, multiply by 100 and attach the percent sign: $\frac{\text{3}}{\text{8}}$ = 0.375 = 37.5%.

Answer: 3/8, 0.375, and 37.5%.

Check. 37.5/100 = 0.375. The shaded amount is less than half, so its percentage should be less than 50%.

Avoid the trap. The bare number 37.5 is not equal to 3/8. A question asking “what percent?” expects the number 37.5; one asking for a probability or proportion expects 0.375 or 3/8.

Recognize the structure. For negative numbers, the value farther left is smaller. Distance is a nonnegative difference in locations.

Work it through. −5/6 = −0.8333... and −4/5 = −0.8. Therefore −5/6 < −0.82 < −4/5. The distance between the fractions is |−$\frac{\text{5}}{\text{6}}$ − (−$\frac{\text{4}}{\text{5}}$)| = |$\frac{\text{−25 + 24}}{\text{30}}$| = $\frac{\text{1}}{\text{30}}$.

Answer: Order: −5/6, −0.82, −4/5; distance: 1/30.

Check. The endpoints are approximately −0.8333 and −0.8000, a gap of about 0.0333.

Avoid the trap. The larger absolute value 5/6 belongs to the smaller negative number. A distance cannot be −1/30.

Key takeaway. The same distinction appears in inequalities, coordinate changes, and absolute-value equations.

Recognize the structure. Multiply the ordinary coefficients and the powers of ten separately. The final coefficient must be at least 1 and less than 10.

Work it through. 4.8(2.5) = 12 and 10⁻⁴10⁶ = 10². Thus

12 × 10² = (1.2 × 10)10² = 1.2 × 10³.

Answer: 1.2 × 10³, or 1,200.

Check. 0.00048 multiplied by 2,500,000 equals 1,200. A tiny factor times a large factor need not be tiny.

Avoid the trap. 12 × 10² is numerically correct but not normalized scientific notation. Changing 12 to 1.2 requires increasing the exponent by 1 to preserve the value.

Key takeaway. Scientific notation makes scale visible and can reduce errors in measurement and conversion problems.

Recognize the structure. The side length is positive and its square is 50. This is a square-root calculation, not division by 2.

Work it through. Let the side length be s. Then s² = 50 and s = $\sqrt{\text{50}}$. Since 7² = 49 and 8² = 64, the length lies between 7 and 8. A calculator gives $\sqrt{\text{50}}$ ≈ 7.0710678, which rounds to 7.1.

Answer: 7.1 meters.

Check. 7.1² = 50.41, close to 50 as expected after rounding. The exact side remains $\sqrt{\text{50}}$, not 7.1.

Avoid the trap. A negative algebraic root is not a physical side length. Also, area has square units, while the side has linear units.

Key takeaway. Use nearby perfect squares to detect bad calculator entries before trusting a radical approximation.

Recognize the structure. Rounding each repeated component changes the total. The exact fractional calculation is easy.

Work it through. Compute the total before rounding:

6 ($\frac{\text{2}}{\text{3}}$) = $\frac{\text{12}}{\text{3}}$ = 4.

To the nearest tenth, the total is 4.0 meters. The student’s early rounding adds 0.7 − 2/3 = 1/30 meter per strip, or 6/30 = 0.2 meter overall.

Answer: 4.0 meters.

Check. Three strips make exactly 2 meters, so six make exactly 4 meters.

Avoid the trap. A rounded component is an approximation, not new exact information. The error can accumulate across many repetitions.

Key takeaway. Retain exact rates and fractions in weighted averages, conversions, and percent calculations until the final answer.

Recognize the structure. The arithmetic combines a small multiplier with division by about one half. An estimate can establish the correct decimal scale before calculation.

Work it through. Use 31.8 ≈ 32, 0.049 ≈ 0.05, and 0.51 ≈ 0.5 to estimate 32(0.05)/0.5 = 3.2. Then calculate without early rounding: 31.8(0.049) = 1.5582, $\frac{\text{1.5582}}{\text{0.51}}$ ≈ 3.0552941.

The thousandths digit is 5, so rounding to hundredths gives 3.06.

Answer: C, 3.06.

Check. The exact calculation is close to the rough estimate 3.2. Choices A, B, and D have the wrong order of magnitude.

Avoid the trap. The estimate identifies the scale, not the exact hundredths. Conversely, exact calculator digits are not trustworthy until the entered expression and decimal scale are checked.

Key takeaway. Combine conceptual checks with computation. Neither mental estimation alone nor unexamined calculator output is a complete solution.