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.

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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.

Mean, Median, Mode, and Weighted MeanRatio and ProportionLinear Equations and ModelingStatistical Graphs and Data Displays

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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.

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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.

Percent and Percent ChangeRatio and ProportionWeighted MeanLinear Equations and Modeling

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