The answer is already on the page. Sometimes it is faster to test it than to find it.
The problem
The sum of three consecutive even integers is 78. What is the largest?
Start in the middle, as always. Try 28 as the largest: the three are 24, 26 and 28, which add to 78. Done in one try. 24 is the smallest of that same triple — the trap for anyone who solves correctly but answers the wrong question.
Three conditions, and all three must hold.
| Condition | Here? |
|---|---|
| the choices are plain numbers | yes |
| the question asks for one specific value | yes — the largest |
| testing a value is quick | yes — one addition |
If all three are true, you never need to set up an equation. You only need to recognise the right answer when you see it.
Answer choices are almost always printed in order — use that.
try C: largest = 28
the middle of the four
the three are 24, 26, 28
count back by 2s
24 + 26 + 28 = 78
exactly the target
If it comes out too big, every larger choice is wrong too — three options die at once. Starting at A can leave you testing all four.
Suppose the sum had come out as 72 instead.
go up — try D
A and B are dead
go down — try B
D is dead
Two tests settle any four-choice question this way. Often one is enough.
A bigger input must give a bigger output. With something like x², where −4 and 4 give the same result, test the choices in order instead.
let the integers be n, n+2, n+4
set up
3n + 6 = 78
collect
n = 24
solve
largest = n + 4 = 28
one more step
n = 24 is choice A, sitting there waiting. Solve properly and forget the last line, and you pick it. Back-solving cannot make that mistake, because you tested the thing the question actually asked for.
Worker P alone needs x hours for a job; worker Q alone needs x + 5 hours. Together they finish in 6 hours. The choices are 6, 8, 10 and 12. Find x.
together per hour: 1x + 1x + 5 should equal 16
rates add, times do not
try 8: 18 + 113 = 21104 ≈ 0.20
16 ≈ 0.17 — too fast
too fast means x is too small → go up
6 dies with 8
try 10: 110 + 115 = 330 + 230 = 530 = 16
exactly the target
A bigger x means slower workers and a longer joint time. So a combined rate that is too big tells you to try a larger x. Decide the direction once, from the story, before you move.
The same move finds a root of a polynomial: for x³ − 2x² − 5x + 6 = 0, test x = 3 — 27 − 18 − 15 + 6 = 0, so 3 is a root. No factoring needed.
A drain works against the taps, so we subtract its rate. Taps that fill in 4 h and 12 h with a drain that empties in 6 h: 14 + 112 − 16 = 312 + 112 − 212 = 212 = 16 of the tank per hour, so the tank fills in 6 hours. To test a choice t, check that t4 + t12 − t6 comes to exactly 1 tank.
A radical equation with a planted false root: √x + 10 = x − 2, choices −1, 3, 6, 10.
try −1: √9 = 3, but −1 − 2 = −3
fails — 3 ≠ −3
try 6: √16 = 4, and 6 − 2 = 4
exactly equal
x = 6
the only true root
What is different: squaring gives x² − 5x − 6 = 0, with roots 6 and −1, and −1 is printed as a choice. Testing in the original equation throws it out automatically.
Choices that are pairs: which (x, y) satisfies 2x + 3y = 12 and x − y = 1? Choices (3, 2), (6, 0), (0, 4), (4, 3).
x − y = 1: (3, 2) ✓ (6, 0) ✗ (0, 4) ✗ (4, 3) ✓
the quicker equation first
2x + 3y: (3, 2) → 12 ✓, (4, 3) → 17 ✗
only two left to test
(3, 2)
passes both
What is different: there is no middle choice to start from. Each pair must pass every equation, and (6, 0) and (0, 4) pass one of them.
The choice is not the unknown: two consecutive odd integers multiply to 143. Their sum is 20, 22, 24 or 26?
sum 22 → 10 and 12
even — impossible, 22 dies
sum 24 → 11 and 13 → 11 × 13 = 143
turn the sum into the pair, then test
sum = 24
what was asked
What is different: each choice must be turned back into the integers before it can be tested — one extra line per test.
Word problems: rebuild the whole story from the choice. Each test needs one fact about the story.
ages — a woman is 30 years older than her son; in 5 years she will be 3 times his age. Try son = 10: now 10 and 40, in 5 years 15 and 45 = 3 × 15 ✓
move both ages by the same years
coins — 18 coins of ₱1 and ₱5 worth ₱50. Try 8 five-peso coins: 8 × 5 = 40, the other 10 coins give ₱10, total ₱50 ✓
count and value must both fit
right triangle — one leg 8, the hypotenuse 4 more than the other leg. Try hypotenuse 10: other leg 6, and 8² + 6² = 64 + 36 = 100 = 10² ✓
leg² + leg² = hypotenuse²
digits — a two-digit number is 7 times the sum of its digits, and swapping the digits makes it 18 smaller. Try 42: 7 × (4 + 2) = 42 ✓, and 24 = 42 − 18 ✓
tens digit a, ones digit b: the number is 10a + b
What is different: the choice is only one piece of the story. We rebuild every other quantity from it, then check every condition — passing one is not enough.
| Question type | Back-solve? | Why |
|---|---|---|
| “find the value of x” | yes | numbers to test |
| age, coin, mixture word problems | yes | testing is one line |
| choices contain letters | no | nothing to substitute |
| “which must be true” | no | one example proves nothing |
| “simplify the expression” | use plugging in | a different technique |
Back-solving tests answers. Plugging in numbers tests expressions. Different tools.
UPCAT-type papers love ages, work rates, “which is a root” and radical equations with four numeric choices — all back-solvable. Start at a middle choice, read the direction from the story, and stop at the first exact fit; one test is often about fifteen seconds.
Back-solve this one
A father is 4 times as old as his son. In 20 years he will be only twice as old. How old is the son now?
Test B = 10: the father is 40 now, and in 20 years they are 60 and 30 — exactly twice. ✓ The algebra agrees: 4s + 20 = 2(s + 20) gives 2s = 20, so s = 10. Back-solving got there without forming the equation at all.
Hint: start at B or C. Multiply by 4 for the father, add 20 to both ages, then see whether the father's new age is exactly double the son's.
Every lesson in the UPCAT Math Review has one of these. 30 questions, marked as you answer, with the working shown for each.
Question 1 of 30
Which value of x satisfies 2^(x) + 2^(x + 1) = 48?
Back-solve: x = 4 gives 16 + 32 = 48. (Algebra: 2^(x)(1 + 2) = 48, so 2^(x) = 16.)
Question 2 of 30
Ana has 22 coins, all ₱5 and ₱10 coins, worth ₱160 in all. How many ₱10 coins does she have?
Back-solve: 10 coins of ₱10 leave 12 coins of ₱5, and 100 + 60 = ₱160. ✓ 12 is the number of ₱5 coins — the right pair, the wrong coin.
Question 3 of 30
Which of the following is a solution of x² − 5x − 14 = 0?
Substitute: 7² − 5(7) − 14 = 49 − 35 − 14 = 0. ✓ −7 gives 49 + 35 − 14 = 70; it comes from flipping the signs in (x − 7)(x + 2).
Question 4 of 30
The sum of three consecutive even integers is 84. What is the product of the smallest and the largest?
84 ÷ 3 = 28 is the middle one, so the integers are 26, 28, 30 and 26 × 30 = 780. 784 = 28² is the trap for using the middle number twice.
Question 5 of 30
Carlo is 6 years older than his sister. Five years ago, he was twice as old as she was. How old is Carlo now?
Test 17: his sister is 11. Five years ago they were 12 and 6, and 12 = 2 × 6. ✓ 11 is the sister's age and 12 is Carlo's age five years ago — both appear in the working but neither is asked for.
Question 6 of 30
Which value of x satisfies √(x + 7) = x − 5?
Test 9: √16 = 4 and 9 − 5 = 4. ✓ Squaring both sides gives x² − 11x + 18 = 0, so x = 2 or 9, but 2 gives √9 = 3 against 2 − 5 = −3 — an extraneous root. Back-solving never picks it up.
Question 7 of 30
Pipe A fills a tank in 6 hours and pipe B fills it in 9 hours, while drain C empties the full tank in 18 hours. If all three are open, how long does it take to fill the empty tank?
Per hour: 1/6 + 1/9 − 1/18 = (3 + 2 − 1)/18 = 4/18 = 2/9 of the tank, so the time is 9/2 = 4.5 hours. Ignoring the drain gives 3.6 h; the answer must be longer than that.
Question 8 of 30
Working together, Ben and Carl paint a room in 4 hours. Alone, Carl takes 6 hours longer than Ben. How many hours does Ben take alone?
Test 6: Carl takes 12, and 1/6 + 1/12 = 2/12 + 1/12 = 3/12 = 1/4 of the room per hour, i.e. 4 hours together. ✓ 12 hours is Carl's time; 4 hours is the time together, and no one alone can be as fast as the pair.
Question 9 of 30
In the right triangle in the figure, the shorter leg is 10 cm and the hypotenuse is 2 cm longer than the other leg. Find the hypotenuse.
(x + 2)² = x² + 10² → 4x + 4 = 100 → x = 24, so the hypotenuse is 26 cm. 24 is the leg, not the quantity asked.
Question 10 of 30
A two-digit number is 4 times the sum of its digits. If 27 is added to the number, its digits are reversed. What is the number?
Test 36: 4(3 + 6) = 36 ✓ and 36 + 27 = 63, the reversal ✓. 24 and 48 also equal 4 times their digit sum, but 24 + 27 = 51 and 48 + 27 = 75 are not reversals — check BOTH conditions. 63 is the reversed number, not the original.
Question 11 of 30
Which value of x satisfies 3(x + 4) = 5x − 6?
Test 9: 3(13) = 39 and 45 − 6 = 39. 5 forgets to multiply the 4 by 3 (3x + 4 = 5x − 6); 3 comes from 2x = 12 − 6.
Question 12 of 30
The sum of four consecutive odd integers is 64. What is the smallest of them?
Try 13: 13 + 15 + 17 + 19 = 64. 19 is the largest of the same four, the answer to a question that was not asked.
Question 13 of 30
Which of the following is a solution of x² + 2x − 35 = 0?
Test 5: 25 + 10 − 35 = 0. −5 gives 25 − 10 − 35 = −20, and 7 gives 49 + 14 − 35 = 28; the true roots are 5 and −7, signs swapped from the trap −5 and 7.
Question 14 of 30
Which value of x satisfies 5^(x) − 5^(x − 1) = 100?
Test 3: 125 − 25 = 100. Test 2: 25 − 5 = 20, too small, so go up; 4 gives 625 − 125 = 500, too big. 25 is the value of 5^(x − 1), not x.
Question 15 of 30
Mara has 20 coins, all ₱5 and ₱10 coins, worth ₱135 in all. How many ₱5 coins does she have?
Try 13 five-peso coins: ₱65, plus 7 ten-peso coins worth ₱70, total ₱135. 7 is the number of ₱10 coins.
Question 16 of 30
A father is 26 years older than his son. In 4 years the father will be 3 times as old as the son. How old is the son now?
Try 9: now 9 and 35; in 4 years 13 and 39, and 39 = 3 × 13. 13 is the son's age in 4 years; 35 is the father's age now.
Question 17 of 30
Which value of x satisfies √(3x + 4) = x − 2?
Test 7: √25 = 5 and 7 − 2 = 5. 0 is the false root from squaring (x² − 7x = 0): √4 = 2 but 0 − 2 = −2. Test in the original equation, not the squared one.
Question 18 of 30
Pipe A fills a tank in 10 hours and pipe B fills it in 15 hours, while drain C empties the full tank in 12 hours. If all three are open, how long does it take to fill the empty tank?
Per hour: 1/10 + 1/15 − 1/12 = 6/60 + 4/60 − 5/60 = 5/60 = 1/12 of the tank, so 12 h. 6 h ignores the drain; 4 h adds the drain's rate instead of subtracting it.
Question 19 of 30
Which pair (x, y) satisfies both 4x + y = 14 and x − 2y = −1?
(3, 2): 12 + 2 = 14 and 3 − 4 = −1, both pass. (2, 6) passes only 4x + y = 14; (5, 3) and (1, 1) pass only x − 2y = −1.
Question 20 of 30
The product of two consecutive positive integers is 182. What is their sum?
27 gives 13 and 14, and 13 × 14 = 182. Two consecutive integers always have an odd sum, so 26 and 28 die at once; 25 gives 12 × 13 = 156.
Question 21 of 30
Working together, Rosa and Ella clean a hall in 8 hours. Alone, Ella takes 12 hours longer than Rosa. How many hours does Rosa take alone?
Test 12: 1/12 + 1/24 = 3/24 = 1/8, exactly the target. 24 is Ella's time, not Rosa's; 8 would make Rosa alone as fast as the pair together.
Question 22 of 30
In a right triangle one leg is 12 cm, and the hypotenuse is 6 cm longer than the other leg. How long is the hypotenuse?
Try 15: the other leg is 9, and 12² + 9² = 144 + 81 = 225 = 15². 13 gives a leg of 7, and 144 + 49 = 193 ≠ 169. 9 cm is the other leg.
Question 23 of 30
Which value of x satisfies √(2x − 1) + 2 = x?
Test 5: √9 + 2 = 5. 1 is the false root: squaring √(2x − 1) = x − 2 gives x² − 6x + 5 = 0 with roots 1 and 5, but √1 + 2 = 3, not 1.
Question 24 of 30
Which of the following is a root of 2x³ − 3x² − 11x + 6 = 0?
Test 3: 54 − 27 − 33 + 6 = 0. −1 gives −2 − 3 + 11 + 6 = 12, 2 gives 16 − 12 − 22 + 6 = −12, and −3 gives −42.
Question 25 of 30
Pipe A fills a tank in x hours and pipe B fills it in 2x hours, while a drain empties the full tank in 6 hours. With all three open the empty tank fills in 12 hours. What is x?
Test 6: 1/6 + 1/12 − 1/6 = 1/12, exactly the target. Test 3: 1/3 + 1/6 − 1/6 = 1/3, far too fast, so x must be larger (bigger x means slower pipes).
Question 26 of 30
A mother is 3 times as old as her son. Eight years ago she was 5 times as old as he was. How old is the son now?
Try 16: now 16 and 48; eight years ago 8 and 40, and 40 = 5 × 8. 8 is the son's age eight years ago. 12 gives 4 and 28, a factor of 7.
Question 27 of 30
The digits of a two-digit number add to 11. Reversing the digits gives a number 45 more than the original. What is the original number?
38: 3 + 8 = 11, and 83 = 38 + 45. 47 reverses to 74, only 27 more; 29 reverses to 92, 63 more. 83 is the reversed number, not the original.
Question 28 of 30
Two consecutive positive odd integers have squares that add to 290. What is the sum of the two integers?
24 gives 11 and 13: 121 + 169 = 290. 22 and 26 split into even pairs (10, 12 and 12, 14), so they die; 20 gives 81 + 121 = 202.
Question 29 of 30
At a school fair, 50 tickets were sold for ₱5,040. Adult tickets cost ₱120 and child tickets ₱80. How many adult tickets were sold?
Try 26 adults: 26 × 120 = ₱3,120, and 24 children give ₱1,920, total ₱5,040. 24 is the number of child tickets.
Question 30 of 30
A rectangle's length is 7 cm more than its width, and its diagonal is 13 cm. What is its area?
60 = 5 × 12, sides 7 apart, and 5² + 12² = 169 = 13². 44 = 4 × 11 gives 16 + 121 = 137, and 78 = 6 × 13 gives 205; neither is 169.
All 176 lessons of the UPCAT Math Review are open to everyone — no account needed. Each comes with its 5-question check. Make a free account and your progress is saved.
The practice set for every lesson and the 7 full-length mock exams come with the UPCAT Math Review — one payment of ₱200, no expiry.
