Showing posts with label UPCAT Mathematics. Show all posts

Thursday, August 20, 2026

Deductive Reasoning UPCAT Reviewer: Conditions, Conclusions, and Quantitative Logic

TEACHER ABI UPCAT MATHEMATICS

Deductive Reasoning

Distinguish what must be true from what may be true using conditions, implication chains, ordering constraints, sets, parity, and counterexamples.

5-10 minute lesson27 original questionsAdaptive practiceSaves progress

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Use only what the statements guarantee

Deductive reasoning begins with given facts and asks what must follow. Do not add assumptions.

If P, then Q: P → Q

You may use the contrapositive:

not Q → not P

But you may not automatically reverse the statement. From P→Q and Q, you cannot conclude P.

Must be true

Follows in every situation satisfying the conditions.

Could be true

Needs only one valid arrangement or example.

Cannot be determined

More than one outcome remains possible.

Contrapositive

P→Q is logically equivalent to not Q→not P.

Counterexample

One valid exception disproves a universal claim.

Ordering chain

Translate comparisons into a single ordered line.

DO IT FAST

Mark facts, links, and gaps

Underline exact facts. Draw arrows for implications and an ordered chain for rankings.

Before selecting an answer, ask:

Does this follow in every valid case?

If you can construct one valid case where the choice fails, it is not a “must be true” conclusion.

Why it works

UPCAT-style reasoning distractors often sound reasonable but reverse a condition, assume overlap between sets, or turn “some” into “all.”

WORKED EXAMPLES

Five forms you should recognize

1. Follow an implication

Problem: All multiples of 8 are even. n is a multiple of 8. What must follow?

multiple of 8 → even

Since n meets the condition, n is even.

2. Do not reverse the arrow

Problem: If a number is divisible by 6, it is divisible by 3. A number is divisible by 3. Must it be divisible by 6?

No. The rule runs from divisible by 6 to divisible by 3, not backward. The number 9 is a counterexample.

3. Build an order

Problem: Lara scored higher than Mina but lower than Nica.

Nica > Lara > Mina

Nica must have the highest score.

4. Use a contrapositive

Problem: If a figure is a square, it has four equal sides. A figure does not have four equal sides.

square → four equal sidesnot four equal sides → not square

The figure cannot be a square.

5. Separate some from all

Problem: Some athletes are musicians. All musicians are readers.

The athletes who are musicians must also be readers, so some athletes are readers. This does not prove that all athletes are readers.

COMMON TRAPS

Check before you commit

Reversing an if-then statement
Treating “some” as “all”
Assuming two groups overlap without evidence
Choosing something merely possible for a must-be-true question
Ignoring the reversal when multiplying an inequality by a negative
Adding real-world assumptions not stated in the problem

FIVE-FORM SKILL CHECK

Do you need the lesson-or just practice?

One original question in each form recommends your next step. It does not yet verify mastery.

CHOOSE YOUR PRACTICE

Work at the level you need.

Foundations

Build the core procedure with immediate explanations.

Core Practice

Use mixed forms with less scaffolding.

UPCAT-Style Transfer

Apply the competency in unfamiliar representations.

FRESH MASTERY CHECK

Ready to verify this competency?

A score of 5/5 verifies mastery. An unsuccessful attempt loads a different five-form bank.

QUICK ANSWERS

Deductive Reasoning FAQ

Is this Mathematics or Language?

It belongs to quantitative reasoning because the logic is applied to numbers, sets, inequalities, arrangements, and formal conditions.

Are the converse and contrapositive the same?

No. The contrapositive of P→Q is not Q→not P and is equivalent; the converse Q→P is not automatically valid.

How do I answer “could be true”?

Test the choices and find one complete arrangement that satisfies every condition.

RELATED COMPETENCIES

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Average Problems UPCAT Reviewer: Missing Values, Combined Groups, and Required Scores

TEACHER ABI UPCAT MATHEMATICS

Average Problems

Use average as a total-value model to recover missing data, combine unequal groups, find required scores, and handle values that enter, leave, or change.

5-10 minute lesson27 original questionsAdaptive practiceSaves progress

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An average represents a total

The arithmetic mean is not just “add and divide.” It connects three quantities:

average = total ÷ number of valuestotal = average × number of values

Most UPCAT-style average problems hide the total. Recover the old total, describe what changed, then divide by the correct new count.

Recover a total

Multiply the average by the number of values.

Missing value

Subtract the known values from the required total.

Combined groups

Add group totals, then divide by the combined count.

Value added or removed

Update both the total and the number of values.

Corrected entry

Change the total by corrected value−recorded value.

Average speed

Use total distance÷total time, not usually the mean of two speeds.

DO IT FAST

Use OLD → CHANGE → NEW

Write three lines:

OLD total = old average × old countNEW total = old total ± changeNEW average = new total ÷ new count

For a required score, compute the target total first and subtract the current total.

Why it works

Averages of averages are not automatically valid. Each group must be weighted by its number of observations.

WORKED EXAMPLES

Five forms you should recognize

1. Find a missing score

Problem: Four quiz scores average 15. Three scores are 12, 14, and 19. Find the fourth.

required total = 4(15) = 60known total = 12 + 14 + 19 = 45missing score = 60−45 = 15

2. Required next score

Problem: Lia’s first four tests average 82. What fifth score gives her an average of 85?

current total = 4(82) = 328target total = 5(85) = 425required score = 425−328 = 97

3. Combine unequal groups

Problem: A class of 20 averages 78 and a class of 30 averages 84. Find the combined average.

combined total = 20(78) + 30(84) = 4080combined average = 4080÷50 = 81.6

Why: The larger class contributes more scores, so 81 is not the answer.

4. Remove a value

Problem: Six measurements average 24. After one is removed, five average 22. Find the removed value.

original total = 6(24) = 144remaining total = 5(22) = 110removed value = 144−110 = 34

5. Average speed

Problem: A vehicle travels 120 km at 60 kilometers per hour and another 120 km at 40 kilometers per hour. Find its average speed.

time = 12060 + 12040 = 2 + 3 = 5 hoursaverage speed = 240÷5 = 48 kilometers per hour

Why: Average speed uses total distance divided by total time.

COMMON TRAPS

Check before you commit

Averaging two group averages without checking group sizes
Using the old count after a value is added or removed
Subtracting a removed value from the average instead of the total
Forgetting to correct the total before dividing
Assuming the required next score equals the desired average
Taking the simple mean of speeds for equal-distance trips

FIVE-FORM SKILL CHECK

Do you need the lesson-or just practice?

One original question in each form recommends your next step. It does not yet verify mastery.

CHOOSE YOUR PRACTICE

Work at the level you need.

Foundations

Build the core procedure with immediate explanations.

Core Practice

Use mixed forms with less scaffolding.

UPCAT-Style Transfer

Apply the competency in unfamiliar representations.

FRESH MASTERY CHECK

Ready to verify this competency?

A score of 5/5 verifies mastery. An unsuccessful attempt loads a different five-form bank.

QUICK ANSWERS

Average Problems FAQ

When may I average two averages directly?

When the two groups have equal numbers of observations.

Why can a required score exceed the target average?

It must compensate for earlier scores below the target.

Is weighted mean part of this competency?

Yes when groups or components contribute unequal amounts; multiply each mean or score by its weight.

RELATED COMPETENCIES

Continue your mathematics review.

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Return to Student Hub View UPCAT Coverage

Age Problems UPCAT Reviewer: Past, Present, and Future Ages

TEACHER ABI UPCAT MATHEMATICS

Age Problems

Translate past, present, and future age relationships into equations and solve multi-condition UPCAT-style age problems.

5-10 minute lesson27 original questionsAdaptive practiceSaves progress

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Put every age on one timeline

Choose one variable for a person’s present age. Express every other age from it.

t years ago: present age − tt years from now: present age + t

If one person is 6 years older, that difference stays 6 in the past and future. A ratio such as 2:1 usually changes because equal years are added to both ages.

Present age

Define variables at the same point in time.

Years ago

Subtract the same number from every person involved.

Years from now

Add the same number to every person involved.

Age difference

The difference between two ages stays constant.

Age ratio

The ratio generally changes over time.

DO IT FAST

Make a three-column timeline

Write three headings: Past | Present | Future. Place each person’s age under the correct heading before forming an equation.

For “in t years, A will be twice B,” write:

A + t = 2(B + t)

Do not write A + t = 2B + t; the time change belongs inside the multiplication.

Why it works

Most age-problem errors come from comparing ages at different times or addingsubtracting years from only one person.

WORKED EXAMPLES

Five forms you should recognize

1. Present ages from a sum

Problem: Mara is 6 years older than Liza. Their present ages total 30. Find Mara’s age.

Setup: Let Liza = x, so Mara = x + 6.

x + (x + 6) = 302x = 24 → x = 12

Mara is 12 + 6 = 18.

2. A future multiple

Problem: A father is 36 and his son is 12. In how many years will the father be twice as old?

Setup: Both ages increase by t.

36 + t = 2(12 + t)36 + t = 24 + 2t → t = 12

Check: In 12 years they will be 48 and 24.

3. A past multiple

Problem: Ana is 24 and Bea is 16. How many years ago was Ana twice Bea’s age?

24−t = 2(16−t)24−t = 32−2t → t = 8

Check: Eight years ago they were 16 and 8.

4. Ratio and difference

Problem: Two siblings’ ages are in the ratio 3:5 and differ by 8 years. Find the older age.

The difference of 2 ratio-parts equals 8, so one part is 4.

older age = 5(4) = 20

5. Two time conditions

Problem: Five years ago, Carlo was half Dana’s age. In 5 years, their ages will total 50.

Let their present ages be C and D.

C−5 = (D−5)2(C + 5) + (D + 5) = 50

The second equation gives C + D = 40. Solving the system gives C = 15 and D = 25.

COMMON TRAPS

Check before you commit

Adding years to only one person
Multiplying before placing both future ages in parentheses
Assuming an age ratio stays constant
Changing the age difference over time
Using “older than twice” as 2(x + k) instead of 2x + k
Comparing one person’s past age with another’s present age

FIVE-FORM SKILL CHECK

Do you need the lesson-or just practice?

One original question in each form recommends your next step. It does not yet verify mastery.

CHOOSE YOUR PRACTICE

Work at the level you need.

Foundations

Build the core procedure with immediate explanations.

Core Practice

Use mixed forms with less scaffolding.

UPCAT-Style Transfer

Apply the competency in unfamiliar representations.

FRESH MASTERY CHECK

Ready to verify this competency?

A score of 5/5 verifies mastery. An unsuccessful attempt loads a different five-form bank.

QUICK ANSWERS

Age Problems FAQ

Does an age difference ever change?

No. If one person is 8 years older now, the difference was and will remain 8 years.

Does an age ratio stay the same?

Usually not. Equal additions change a ratio unless the ages are equal.

How do I check my answer?

Substitute the present ages into every past or future condition in the problem.

RELATED COMPETENCIES

Continue your mathematics review.

SAVE AND CONTINUE

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Mastery results save to your Teacher Abi study profile.

Return to Student Hub View UPCAT Coverage

Tuesday, August 18, 2026

Mixture Problems UPCAT Reviewer: Concentration, Blending, and Replacement

TEACHER ABI UPCAT MATHEMATICS

Mixture Problems

Model mixture problems by tracking the conserved substance or total value, then distinguish dilution, evaporation, blending, and replacement situations.

5-10 minute lesson27 original questionsAdaptive practiceSaves progress

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Track the amount that is being conserved

This is a Mathematics modeling topic. The liquid, alloy, food, or price context supplies quantities for a weighted equation; no Chemistry knowledge is required.

For concentration problems, track the amount of pure substance:

pure amount = concentration × total amount

For price blends, track total value:

total value = unit price × quantity

Then write:

amount before + amount added − amount removed = amount after

Concentration

Write a percent as a decimal before multiplying.

Dilution

Adding water changes total volume but not pure solute.

Evaporation

Only water leaves; the pure solute remains.

Price blend

Equate the sum of ingredient values to the blend’s total value.

Replacement

Removing mixture removes every component in the same current proportion.

DO IT FAST

Build a quantity–rate–pure amount table

For every ingredient, record:

quantity × concentration = pure amount

Add the pure-amount column, not the concentration column. If the total mixture is fixed, express one quantity as total−x.

For repeated replacement, multiply by the fraction retained at every step.

Why it works

Concentrations cannot usually be averaged directly. The conserved pure amount or total value determines the correct weighted result.

WORKED EXAMPLES

Five forms you should recognize

1. Combine two solutions

Problem: Two liters of a 20% salt solution are mixed with 3 liters of a 40% solution. Find the final concentration.

Setup: Add the pure salt, then divide by the combined volume.

0.20(2) + 0.40(3) = 1.6 L salt1.6÷(2 + 3) = 0.32 = 32%

Why: Concentrations are weighted by their quantities.

2. Dilute with water

Problem: How much water must be added to 10 L of a 30% solution to make it 20%?

Setup: The solution initially contains 0.30(10) = 3 L of pure substance. Water changes the total volume, not that 3 L.

final total = 3÷0.20 = 15 Lwater added = 15−10 = 5 L

3. Blend prices

Problem: A merchant has 20 kg of rice worth ₱80 per kilogram. How many kilograms worth ₱140 per kilogram must be added to produce a blend worth ₱100 per kilogram?

Setup: Total ingredient value equals total blend value.

80(20) + 140x = 100(20 + x)1600 + 140x = 2000 + 100x → x = 10 kg

4. Evaporation

Problem: A 25 L syrup mixture is 12% sugar. How much water must evaporate for it to become 15% sugar?

Setup: The 3 L of sugar remains; only water leaves.

0.12(25) = 3 L sugarfinal total = 3÷0.15 = 20 Lwater evaporated = 25−20 = 5 L

5. Remove and replace

Problem: A 20 L container holds a 30% salt solution. Five liters are removed and replaced with water. Find the new concentration.

Setup: Removing 5 of 20 liters removes one-fourth of the salt.

salt remaining = 0.30(20)(1520) = 4.5 Lnew concentration = 4.5÷20 = 22.5%

Why: Replacement restores the total volume but adds no salt.

COMMON TRAPS

Check before you commit

Averaging concentrations without considering quantities
Adding water to the pure-solute amount
Treating evaporation as removal of the whole mixture
Forgetting that removed mixture contains solute
Using the original concentration during a second replacement step
Mixing pesos per kilogram with total pesos

FIVE-FORM SKILL CHECK

Do you need the lesson-or just practice?

One original question in each form recommends your next step. It does not yet verify mastery.

CHOOSE YOUR PRACTICE

Work at the level you need.

Foundations

Build the core procedure with immediate explanations.

Core Practice

Use mixed forms with less scaffolding.

UPCAT-Style Transfer

Apply the competency in unfamiliar representations.

FRESH MASTERY CHECK

Ready to verify this competency?

A score of 5/5 verifies mastery. An unsuccessful attempt loads a different five-form bank.

QUICK ANSWERS

Mixture Problems FAQ

When can I simply average two concentrations?

Only when equal quantities are mixed. Otherwise use a weighted average based on quantity.

What remains constant during dilution?

The amount of pure solute remains constant; total volume changes.

Why does repeated replacement use multiplication?

Each step retains a fraction of the amount present immediately before that step.

RELATED COMPETENCIES

Continue your mathematics review.

SAVE AND CONTINUE

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Mastery results save to your Teacher Abi study profile.

Return to Student Hub View UPCAT Coverage

Monday, August 17, 2026

Pattern Recognition UPCAT Reviewer: Number, Figure, and Cyclic Patterns

TEACHER ABI UPCAT MATHEMATICS

Pattern Recognition and Quantitative Sequences

Recognize whether a pattern changes by position, alternates between rules, grows through differences, or repeats in a cycle—and verify the rule against every given term.

5-10 minute lesson27 original questionsAdaptive practiceSaves progress

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Look for structure before choosing a formula

A pattern may use one repeated operation, changing differences, alternating rules, two interleaved sequences, a recurrence, or a repeating cycle.

Do not accept a rule merely because it produces the next term. A valid rule must explain all the given terms.

First differences

Subtract consecutive terms; constant differences indicate linear growth.

Second differences

Differences of the differences can reveal square or quadratic growth.

Alternating rule

Odd and even positions may follow different operations or sequences.

Recurrence

Each term is produced from one or more earlier terms.

Cycle

Last digits and remainders often repeat after a fixed number of steps.

DO IT FAST

Use the difference–ratio–position scan

1. Differences: subtract neighboring terms.

2. Ratios: divide neighboring terms if multiplication seems likely.

3. Positions: separate odd- and even-position terms.

4. Recurrence: test operations such as “double then add.”

5. Cycle: list one complete repeating block, then use the position’s remainder.

Why it works

This prevents forcing every sequence into arithmetic or geometric form and helps you detect the short structural rule UPCAT-style items are designed around.

WORKED EXAMPLES

Five forms you should recognize

1. Changing differences

3, 7, 13, 21

differences: 4, 6, 8 → next difference 10

2. Alternating operations

2, 4, 5, 10, 11

×2, +1, ×2, +1 → next ×2

3. Interleaved sequences

2, 10, 4, 20, 6, 30

Odd positions count by 2; even positions count by 10.

4. Visual growth

Adjacent squares use 4, 7, 10, … matchsticks because each new square shares one side and adds only three.

5. Cyclic remainder

Units digits of powers of 7 repeat 7, 9, 3, 1. Reduce a large exponent by the four-term cycle.

COMMON TRAPS

Check before you commit

Checking only the last two terms
Assuming every pattern has constant differences
Ignoring separate odd- and even-position rules
Using a formula that fails an earlier term
Forgetting shared sides in growing figures
Using remainder 0 as the first rather than last position of a cycle

FIVE-FORM SKILL CHECK

Do you need the lesson-or just practice?

One original question in each form recommends your next step. It does not yet verify mastery.

CHOOSE YOUR PRACTICE

Work at the level you need.

Foundations

Build the core procedure with immediate explanations.

Core Practice

Use mixed forms with less scaffolding.

UPCAT-Style Transfer

Apply the competency in unfamiliar representations.

FRESH MASTERY CHECK

Ready to verify this competency?

A score of 5/5 verifies mastery. An unsuccessful attempt loads a different five-form bank.

QUICK ANSWERS

Pattern Recognition and Quantitative Sequences FAQ

Can more than one rule fit a short sequence?

Yes. Standardized-test items assume the simplest consistent rule supported by the answer choices and all supplied terms.

Is pattern recognition the same as arithmetic sequences?

Arithmetic sequences are one type. Pattern recognition also includes changing differences, alternation, recurrence, cycles, and visual growth.

How do I handle a huge exponent?

Find the repeating last-digit or remainder cycle, then locate the exponent’s position within that cycle.

RELATED COMPETENCIES

Continue your mathematics review.

SAVE AND CONTINUE

Your progress stays on this browser.

Mastery results save to your Teacher Abi study profile.

Return to Student Hub View UPCAT Coverage