TSIA2 study guide

The quantitative diagnostic includes basic operations, rounding, and comparing numbers written in different forms. Translate the task before calculating, use one common form when comparing, and keep exact intermediate values until a rounding instruction applies.

1. Keep the order of a subtraction. “Six less than a total” means total minus six. Subtracting a negative is adding its opposite; for example, 4 − (−9) = 13.
2. Apply parentheses before multiplication and division, and those operations before addition and subtraction. Operations at the same priority are performed from left to right.
3. Add or subtract fractions using a common denominator. Multiply numerators and denominators when multiplying. Divide by a nonzero fraction by multiplying by its reciprocal.
4. For decimal addition and subtraction, align the decimal points. For multiplication, track place value rather than assuming a longer decimal string means a larger value.
5. Convert comparison values to a common form. For instance, 3/8 = 0.375 = 37.5%. With negative values, the one closer to zero is greater.
6. Round directly to the requested place by looking at the next digit. State how halfway values are handled when it matters. Repeated rounding can produce a different result from one direct rounding.
7. Check the requested quantity and unit. A remaining balance requires subtraction; a count of whole purchases may require rounding down or up according to the situation.

Common mistake: Adding fraction denominators, dropping a negative sign, or rounding an intermediate value can change the result. A decimal ending in extra zeros is not automatically larger: 0.6 and 0.600 are equal.

Practice tip: Write a short exact calculation first. Use common denominators or a common decimal form, then apply the requested rounding only at the end.

A ratio compares quantities; a proportion preserves their relative sizes. A percent is a ratio with a base of 100. Before calculating, identify the comparison order and the whole or original amount that acts as the base.

1. Label the quantities in a ratio. A part-to-part ratio of 3:5 has 8 total parts, so the first part is 3/8 of the whole, not 3/5.
2. For proportional quantities, multiply or divide both by the same scale factor, or find the constant quotient between corresponding values.
3. For a percent problem, use part = rate × whole with the rate in decimal or fraction form. Divide the part by the rate to recover an unknown whole.
4. Represent a percent increase by 1 + rate and a percent decrease by 1 − rate. Apply successive changes to the current value at each step.
5. When reversing changes, undo them in reverse order by dividing by their multipliers. Do not reverse a discount by adding the same percent.
6. Check the base: percentage points subtract two percentages, while relative percent change divides their difference by the original percentage.

Common mistake: Do not assume equal percent increases and decreases cancel or reverse one another. The base changes after the first adjustment. Also distinguish a ratio of two parts from a fraction of the whole: concentrate:water = 2:7 means concentrate is 2/9 of the mixture.

Practice tip: Write a short label for every denominator or percent base. For a multistep percent calculation, record each multiplier and check the final answer by applying the original changes in order.

A square-root symbol can represent an exact value even when its decimal expansion never ends. Compare signs and known squares before relying on a rounded display. Bounds are often enough to answer a magnitude question exactly.

1. A rational number can be written as a ratio of integers with a nonzero denominator. Terminating and repeating decimals are rational. Square roots of positive integers that are not perfect squares are irrational.
2. For nonnegative quantities, comparing squares preserves order. Because 16 < 18 < 25, the positive square root of 18 lies between 4 and 5.
3. Keep exact operations separate from estimates. For nonnegative a and b, √a × √b = √(ab); this does not mean √a + √b = √(a + b).
4. An irrational number plus a rational number is irrational. Two irrational numbers can add or multiply to a rational result, so examine the actual expression.
5. For positive denominators, the smaller denominator produces the larger reciprocal. For negative values, reverse the intuition based on magnitude: −4.1 is less than −3.9.
6. Propagate a bound through each operation. Multiplying by a positive number preserves its order. Subtracting the bounded quantity from a fixed value reverses which endpoint gives the smaller result.
7. A finite calculator display is a rational approximation. Keep the distinction between equality and approximation, especially when the choices are close.

Common mistake: Squaring preserves order only when the compared values are known to be nonnegative. Also, √a + √b generally cannot be combined into one root by adding a and b.

Practice tip: Start with the sign, then look for nearby perfect squares or convenient rational bounds. Use a calculator to check the size when available, while keeping the exact reasoning clear.

A linear expression combines constant rates times quantities with fixed amounts. The quantitative strand asks you to identify, transform, and interpret these forms. Keep each variable’s meaning and each amount’s unit visible as you rewrite.

1. Define each variable before translating. If n counts items, 6n is a repeated $6 charge per item, while +14 is a fixed $14 charge added once.
2. Use parentheses for a change applied to a whole subtotal. A 9% charge on S is 0.09S, and the subtotal plus the charge is 1.09S.
3. Distribute a factor to every term, including negative signs: −3(a − 2) = −3a + 6. Combine terms only when their variable parts match.
4. Equivalent forms have the same value for every allowed input. Factoring reverses distribution and can reveal a common unit or a repeated group.
5. Translate limits precisely: at most uses ≤, less than uses <, and at least uses ≥. Multiplying both sides of an inequality by a negative reverses its direction.
6. Interpret a coefficient using the output and input units. In dollars = 0.28 × pages + 11, the rate is 28 cents per page and the fixed amount is $11.
7. After rewriting, connect each term to the situation. If a variable cancels completely, the resulting quantity does not depend on that variable.

Common mistake: A fixed charge is not repeated for every item, and a discount or fee may apply to only part of a bill. Read the scope of the percentage before choosing parentheses.

Practice tip: Annotate each term with its role—rate, quantity, fixed charge, or adjustment. Expand or factor once, then check that the new form preserves those roles.

A linear equation describes a balance between quantities. A system asks for values that satisfy every equation at once. The same solution may appear as matching outputs in a table, the intersection of graphs, or numbers obtained by substitution or elimination.

1. Define each variable with its unit. Translate total counts, total costs, and limits separately before combining them.
2. For substitution, isolate one variable and replace it in another equation. For elimination, multiply entire equations as needed, then add or subtract to cancel a variable.
3. Apply reversible operations to both sides. Dividing by a variable expression may discard the case where that expression is zero, so check that case first.
4. Interpret special outcomes: a false constant statement means no solution; an identity from dependent equations means a family of solutions, subject to any remaining constraints.
5. For inequalities, reverse the comparison when multiplying or dividing by a negative. AND requires both conditions; OR permits either. Strict inequalities exclude their boundary.
6. Substitute the solution into the original equations, then verify whole-number, nonnegative, or budget requirements imposed by the situation.

Common mistake: A pair that satisfies only one equation is not a system solution. Also, a larger raw count need not fit a cost limit, and an algebraic decimal count may be impossible when the objects must be whole.

Practice tip: Keep one equation for each independent condition, preserve the units, and finish by checking every original condition rather than only your last simplified equation.

Simplification changes the form of an expression while preserving its values on the original allowed inputs. Factoring exposes shared factors, but cancellation never makes a previously undefined input valid. Radicals and rational powers require special attention to real-number restrictions.

1. Record the original restrictions before simplifying: every denominator and every divisor must be nonzero; even-root radicands must be nonnegative.
2. Factor whole expressions before canceling. Only common multiplicative factors cancel; terms joined by addition or subtraction do not cancel individually.
3. Use a common denominator to add rational expressions. When dividing by a rational expression, also exclude inputs where that divisor is zero before multiplying by its reciprocal.
4. Combine polynomial terms only when their variable powers match. In products, distribute to every term, and reverse every sign when subtracting a polynomial.
5. For positive bases, add exponents in products, subtract in quotients, and multiply for powers of powers. For a real variable, √(u²) equals |u|, not always u.
6. Check equivalence at several allowed inputs as an error check, and retain every original exclusion in the final statement. Substitution alone is not a proof for all inputs.

Common mistake: Writing only the restrictions visible in the final denominator loses holes created by cancellation. Dividing by an expression also creates restrictions that are easy to miss if you check only printed denominator bars.

Practice tip: Write the excluded values beside the original expression before doing any cancellation. Carry that list through every subsequent line.

A model should reflect how its quantities interact. Multiplying two changing linear quantities often creates a quadratic. Repeating a percentage change creates an exponential. The equation, table, and graph each reveal useful features when their inputs and outputs are kept clear.

1. For quadratic contexts, identify the product that creates a square term, such as price times demand or one rectangle side times another. Define the physically allowed input interval.
2. Use factored form to find zeros and vertex form to identify the maximum or minimum. A negative leading coefficient gives a maximum; a positive leading coefficient gives a minimum.
3. For exponential change, write initial amount × (factor)^(number of periods). An r-percent increase uses factor 1 + r/100; a decrease uses 1 − r/100.
4. Convert elapsed time to the number of growth periods. Doubling every three years uses exponent t/3 when t is in years, not 3t.
5. Reverse exponential growth by dividing by the accumulated factor. Consecutive percentage changes multiply; their percentages generally do not simply add.
6. After solving, enforce context restrictions and requested precision. For a first whole period above a threshold, check the preceding period as well.

Common mistake: An x-intercept gives a zero output, not the maximizing input. In exponential models, a 15% loss leaves a factor of 0.85, and undoing a 15% gain requires division by 1.15 rather than subtracting 15%.

Practice tip: Name the feature requested before choosing a method: a zero, a maximum, a growth factor, an initial amount, or a first threshold crossing. Then interpret the mathematical result in the original units.

Length measures one dimension, area measures a surface, and volume measures three-dimensional space. Match all units before combining measurements. A length conversion factor must be squared for area and cubed for volume.

1. Identify the quantity: length, area, volume, time, or temperature. This determines what kind of unit the answer needs.
2. Choose one consistent unit for each dimension. Write proportional conversion factors as ratios so the old units cancel. Absolute temperatures can require a formula with an offset, as in Celsius-to-Fahrenheit conversion.
3. Square a length factor for square units and cube it for cubic units. For example, 1 m² = 10,000 cm² and 1 m³ = 1,000,000 cm³.
4. Calculate with the converted measurements, then check that the resulting unit has the right dimension.

Common mistake: Using 18 as the depth while the other dimensions are in meters produces a volume 100 times too large. Converting square or cubic units with only the linear conversion factor is another common error.

Practice tip: Write a unit beside each intermediate result. Units that do not cancel as intended often reveal the incorrect operation before you finish.

A combined figure can often be measured by adding simple pieces or subtracting a missing region from a larger shape. Area, volume, and perimeter require different bookkeeping: an internal seam contributes no outside perimeter, but each component still contributes its volume.

1. Sketch the figure and label the dimensions that belong to each component.
2. Choose formulas for the quantity requested. For volume, identify each base area and perpendicular height.
3. Add non-overlapping component measures, or subtract holes and cutouts. Count only exposed edges or faces when finding outside perimeter or surface area.
4. Keep exact values such as multiples of π until the final requested approximation.

Common mistake: Using the cone’s slant height instead of its perpendicular height gives the wrong volume. For a surface-area question, adding both complete component surfaces would also incorrectly count their hidden shared face.

Practice tip: State what each intermediate number measures. A circular base area, a curved surface area, and a solid volume can involve similar numbers but have different units.

Translations, rotations, and reflections preserve lengths and angles. A dilation with a positive scale factor multiplies corresponding lengths by that factor and preserves angle measures. Areas scale by the square of that factor and volumes by its cube.

1. Identify whether the operation is a rigid motion or a dilation. Check any stated center, line, and direction.
2. For a dilation centered at C, find the displacement from C to the original point P.
3. Multiply that displacement by the scale factor and add the center back: P′ = C + k(P − C).
4. For a positive scale factor k, use k for lengths, k² for areas, and k³ for volumes. For a sequence of transformations, apply them in the stated order.

Common mistake: Multiplying the original coordinates directly assumes the center is the origin. Multiplying the area by only 1.5 also misses the second dimension of the enlargement.

Practice tip: Check that the center stays fixed and that the original and image points lie on the same ray from the center when the scale factor is positive.

Horizontal and vertical coordinate differences form perpendicular legs of a right triangle. The Pythagorean theorem gives distance. Relative to a chosen acute angle, sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent.

1. Draw the right triangle and identify its right angle. The opposite side is the hypotenuse.
2. When coordinates are given, subtract x-coordinates and y-coordinates to find the horizontal and vertical changes.
3. Use a² + b² = c² for side lengths. Choose sine, cosine, or tangent according to the sides involved and the location of the requested angle.
4. Check that the hypotenuse is the longest side and that a sine or cosine ratio for an acute angle lies between 0 and 1.

Common mistake: Using 8 and 10 as the legs measures from the origin, not from A. The required legs are coordinate differences. Reversing the tangent ratio gives the tangent of the other acute angle.

Practice tip: Mark the chosen angle before labeling opposite and adjacent. Those labels change when you switch angles, even though the triangle stays the same.

A probability compares favorable outcomes with a clearly defined set of possible outcomes. The denominator changes when information restricts the group or a draw removes an object. Words such as both, or, given, exactly, and at least describe different events; translate them before selecting an operation.

1. For equally likely outcomes, use favorable outcomes divided by all possible outcomes. For observed data, use the event frequency divided by the number of recorded trials. A fraction between 0 and 1 is required.
2. The complement of an event contains all outcomes in which it does not occur. Its probability is 1 minus the event probability. For “at least one,” it is often easier to count “none” and subtract from 1.
3. For “A or B,” include outcomes in either event. If the groups overlap, subtract their intersection once to avoid double-counting: P(A or B) = P(A) + P(B) − P(A and B).
4. For “A and then B,” multiply the chance of A by the chance of B under the conditions left after A. Independent events keep the same second probability. Without replacement, update both the remaining favorable count and the total.
5. For “given B,” restrict the sample space to B before counting. P(A given B) usually differs from P(B given A). A two-way table helps identify the correct row, column, or restricted total.

Common mistake: Adding three copies of 4/9 ignores the overlap among successes and the effect of removing buttons. Keeping 9 as every denominator also treats earlier draws as if they were replaced.

Practice tip: Write the event in words and list the allowed outcomes before doing arithmetic. Check whether the two stages are independent and whether a condition has narrowed the denominator.

A center describes a typical location in a data set; a spread describes how dispersed its values are. Neither alone describes a distribution fully. Means depend on every numerical value, while medians depend mainly on position after sorting. Always name the measure and the convention used.

1. The mean is the sum divided by the observation count. A frequency table requires multiplying each value by its frequency before adding. To combine groups, recover each group’s total from mean × count, then divide the combined total by the combined count.
2. Sort before finding a median. With an odd count, select the middle observation; with an even count, average the two central observations. Keep duplicate values because each represents an observation.
3. Range is maximum minus minimum. Interquartile range is Q3 minus Q1 and describes the width of the middle half. If a problem specifies a quartile convention, follow it consistently rather than relying on a calculator’s default.
4. An unusually large or small observation can move the mean substantially. The median and IQR are usually more resistant. A common outlier rule flags values below Q1 − 1.5(IQR) or above Q3 + 1.5(IQR), with strict inequalities.
5. A constant added to every value changes the mean and median by that constant but leaves spreads unchanged. Multiplying every value by a positive factor scales the center and spreads by that factor. Standard deviation summarizes distances from the mean and uses the original measurement units.

Common mistake: Averaging subgroup means without their group sizes gives the wrong combined mean when the groups have different counts. Another common error is using the full range in an IQR outlier rule.

Practice tip: Write the ordered data or the frequency-weighted totals. State whether the result describes center or spread, and check that the final unit matches the measurements.

A display can summarize recorded values well while still supporting only a limited claim. Separate the type of variable, the way observations were selected, and the way any treatment was assigned. Good interpretation uses the information actually available and preserves the relevant denominators.

1. Categorical values name groups; quantitative values measure or count amounts. Numerical labels such as room numbers can still be categorical. Counts are discrete, while measurements such as mass are usually modeled as continuous.
2. Use bars for category frequencies, histograms or dot plots for one numerical distribution, and scatterplots for paired numerical measurements. A line graph keeps time order. Side-by-side box plots compare centers and spreads on a common scale.
3. Read units, scales, group totals, and interval boundaries. Raw counts can mislead when group sizes differ; compare rates using each group’s own denominator. A histogram interval does not reveal exact values within it. A shortened bar-chart baseline can exaggerate a visual ratio.
4. Random sampling helps a sample represent its target population. Convenience samples, voluntary response, undercoverage, and nonresponse can introduce bias. Increasing the sample size alone does not remove those biases.
5. Random assignment helps support a causal treatment comparison. An observational association can have a confounding cause or a reverse direction. Predictions far beyond the observed input range are extrapolations and require additional assumptions.

Common mistake: Random selection into a sample and random assignment to a treatment solve different problems. One improves population representation; the other helps separate treatment effects from preexisting differences.

Practice tip: Before accepting a claim, ask what was measured, who was observed, whether treatments were assigned, and whether the statement concerns a group average or every individual.