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UPCAT Math Review · Word problems

Two Investments: Track the Interest, Not the Money

Split money earns split interest. Let x be one share, total − x the other — then write the equation about the INTEREST, because that is the number the story gives you.

The problem

₱50,000 is split between accounts earning 4% and 6% a year. Total interest for the year is ₱2,600. How much was invested at 6%?

One variable covers both shares.

at 6%: x · at 4%: 50,000 − x

the two shares must total the money

Resist inventing a second variable. "Total − x" already says everything the second account knows, and it keeps the problem one equation deep.

The equation lives in the interest column.

0.06x + 0.04(50,000 − x) = 2,600

each share times its own rate

2,000 + 0.02x = 2,600

expand and collect

x = ₱30,000 at 6%

and ₱20,000 at 4%

When one interest is ₱N more than the other, subtract instead of add. ₱20,000 is split between 9% and 3%, and the 9% part earns ₱600 more: 0.09x − 0.03(20,000 − x) = 600, so 0.12x − 600 = 600, 0.12x = 1,200 and x = ₱10,000 at 9% (check: 900 − 300 = 600).

A sanity bracket before solving anything.

all at 4% → ₱2,000 · all at 6% → ₱3,000

the interest must land between these

2,600 sits 60% of the way up

so ~60% of the money sits at 6% — 30,000 ✓

That bracket is a free answer-checker: any choice implying interest outside ₱2,000–₱3,000 is dead before the algebra starts.

The recurring traps.

Trap	What goes wrong
answering the other account	both shares appear among the choices — reread which was asked
rates on the wrong shares	0.04x when x was defined at 6%
percent left as a whole number	6x instead of 0.06x — interest a hundredfold too big

Amounts linked without a total.

Let x be the smaller share; build the other from it.

x at 3%, x + 5,000 at 7%

one letter

0.03x + 0.07(x + 5,000) = 1,350

interest column

0.10x + 350 = 1,350

x = 10,000

the 3% share

total = 10,000 + 15,000 = ₱25,000

the question asked for all of it

Answer the amount asked

₱10,000 and ₱15,000 are the two shares; the question wanted the total. All three sit among the choices.

Adding money to reach a blended rate.

The new money counts in the total too.

270 + 0.08y = 0.05(9,000 + y)

0.03 × 9,000 = 270

270 + 0.08y = 450 + 0.05y

0.03y = 180, so y = ₱6,000

check: 270 + 480 = 750 = 5% of 15,000 ✓

Variations: more than one year, a decimal rate, a loss.

Still rate × amount in the interest column; the column just looks different.

Two years, not one: ₱24,000 is split between 5% and 8% simple interest. After 2 years the total interest is ₱3,000. How much is at 5%?

per year: 3,000 ÷ 2 = 1,500

simple interest is the same each year

0.05x + 0.08(24,000 − x) = 1,500

x at 5%

1,920 − 0.03x = 1,500, so x = ₱14,000

and ₱10,000 at 8%

Check: 700 + 800 = ₱1,500 a year, ₱3,000 in two ✓. Interest is rate × amount × years.

Forgetting the 2 years

Setting one year’s interest equal to ₱3,000 gives 1,920 − 0.03x = 3,000 and a negative x. The bracket test catches it: one year can earn at most 8% of 24,000 = ₱1,920.

A decimal rate and equal interests: ₱34,000 is split between 4.5% and 4% so that both parts earn the same interest. How much is at 4.5%?

0.045x = 0.04(34,000 − x)

4.5% = 0.045, not 0.45

0.085x = 1,360, so x = ₱16,000

and ₱18,000 at 4%

Check: 720 = 720 ✓. For equal interest, the higher rate gets the smaller share.

A loss: ₱25,000 is split. One part gains 10% and the other loses 4%. The net gain is ₱1,450. How much was in the part that gained?

0.10x − 0.04(25,000 − x) = 1,450

a loss enters with a minus

0.14x − 1,000 = 1,450

−0.04 × −x = +0.04x

x = 2,450 ÷ 0.14 = ₱17,500

check: 1,750 − 300 = 1,450 ✓

A losing share is just a negative rate. Watch the sign when the bracket is expanded.

Swapped amounts.

Swapping moves 5% onto one share and off the other.

x at 4%, 30,000 − x at 9%

0.05x − 0.05(30,000 − x) = 500

gain on x, loss on the rest

0.10x − 1,500 = 500

x = ₱20,000

Check: before, 800 + 900 = ₱1,700; after the swap, 1,800 + 400 = ₱2,200 — ₱500 more ✓.

On the exam

Investment items put the equation in the interest column: rate × amount for each share. Check that the total interest lies between “all at the low rate” and “all at the high rate”, and answer the share (or the total) the question asks for.

Check: the two interests rebuild the total.

6% share 0.06 × 30,000 = 1,800 the bigger earner

4% share 0.04 × 20,000 = 800 1,800 + 800 = 2,600 ✓

Remember the flow

x and total − x · Equation on the interest · Bracket, solve, reread the question

Both shares appear in the choices — answer the one that was asked.

Now you try

Practice problem

The problem

₱20,000 is split between 3% and 5% accounts so that each earns the same interest. How much sits at 3%?

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