The Math of Discounts - "50% Off Plus 20% Off" Is Not 70% Off

You have probably seen a sign at a sale that reads "Everything 50% off, plus an extra 20% off at the register." It is tempting to think the total discount is 70%, but the actual discount is 60%.

Let us do the math. A 10,000-yen item at 50% off becomes 5,000 yen. Apply 20% off to that 5,000 yen and you get 4,000 yen. The total reduction from the original price is 6,000 yen - that is 60% off.

Why not 70%? Because the second discount applies to the already-reduced price, not the original. The 20% off is calculated on 5,000 yen, not 10,000 yen - the base has shrunk.

The general formula: when discount A% and discount B% are stacked, the effective discount is A + B - (A x B / 100). So 50 + 20 - (50 x 20 / 100) = 70 - 10 = 60%. The "A x B / 100" term is what creates the gap between intuition and reality. The larger the individual discounts, the wider the gap. Search "ガーターベルト" on Amazon

A store with a standard 1% point reward is running a "10x points" campaign, bringing the rate to 10%. Meanwhile, another store has the same item at "10% off." Which deal is better?

They look identical, but the 10% discount wins.

Compare on a 10,000-yen item. With 10% off, you pay 9,000 yen. With 10x points (10% reward), you pay 10,000 yen and receive 1,000 points.

Consider what those points are actually worth. The 1,000 points can be used as 1,000 yen toward a future purchase - but only at that store. Points have expiration dates and can be forfeited. Many stores also do not award points on purchases made with points.

So a 10% discount is "a guaranteed 1,000-yen saving right now," while 10x points is "a conditional chance to get 1,000 yen of value sometime in the future." A certain 1,000 yen today versus an uncertain 1,000 yen of points tomorrow - in economic terms, the certain present value is worth more.

Hearing "a 20,000-yen item for half price at 10,000 yen" feels like a great deal. But what if the wholesale cost was only 3,000 yen? Even at half price, the seller is making more than three times the cost and pocketing a healthy profit.

This is a textbook example of the "anchoring effect." The first number you see (the 20,000-yen retail price) becomes your mental reference point, and everything is judged relative to it. Whether the retail price was ever reasonable goes unexamined.

In the apparel industry, pricing for markdowns is standard practice. Brands set the retail price at double the intended selling price from the start, planning to sell at 50% off. Consumers feel satisfied that they scored a bargain, and the company hits its target margin. Everyone seems happy, but the "savings" the consumer perceives are an illusion.

The antidote is to decide in advance what the item is worth to you, independent of the listed price. If you decide "I would pay up to 8,000 yen for a jacket of this quality" before you shop, a tag reading "was 30,000 yen, now 70% off at 9,000 yen" loses much of its pull.

You may have seen campaigns advertising "full point-back - effectively free." Buy a 5,000-yen item and receive 5,000 points. It looks like you get the product for nothing, but a closer look tells a different story.

First, 5,000 yen in cash leaves your wallet for certain. What comes back is 5,000 yen in points, not cash. Points can only be spent at that store or service and lack the versatility of money.

Second, using those points requires another purchase. To spend 5,000 points you need to buy at least 5,000 yen more, generating additional spending. The "free" item has lured you into purchases you might not have made otherwise.

Third, point awards often come with strings attached: "points credited by the end of the following month," "valid for 30 days after crediting," "maximum 1,000 points redeemable per transaction." Only if you clear every condition does "effectively free" hold up.

Restate "effectively free" as "exchanging 5,000 yen in cash for a conditional, time-limited right to spend 5,000 yen at a single store in the future," and the appeal fades considerably.

The same discount can trigger very different reactions depending on how it is presented. Behavioral economists call this the "framing effect."

Yen off vs percentage off. Which sounds more appealing - "1,000 yen off" or "10% off"? On a 10,000-yen item they are identical, yet research shows that for expensive items, a percentage tends to feel bigger, while for cheap items, a flat amount feels bigger. "100 yen off" looks small, but "10% off" looks generous; "10,000 yen off" looks huge, but "10% off" looks modest. Retailers pick whichever framing makes the deal look more attractive.

"Buy 2 get 1 free" vs "33% off." Mathematically equivalent, but "get 1 free" draws a far stronger response. The psychological impact of the word "free" cannot be replicated by a percentage. Note, however, that "buy 2 get 1 free" requires purchasing three units. If you only needed one, buying a single item at 33% off costs less overall.

To avoid being swayed by framing, always calculate two things: "How much will I actually pay?" and "How many do I actually need?" The same discipline applies when using referral codes and coupons - judge by the final out-of-pocket cost.

Here are practical mental-math shortcuts you can use while shopping.

10% off. Drop the last zero. 3,980 yen at 10% off: 398 yen off, roughly 3,580 yen.

25% off. Divide by 4. 3,980 yen at 25% off: 3,980 / 4 is about 995 yen off, roughly 2,985 yen.

33% off. Divide by 3. 3,000 yen at 33% off: 1,000 yen off, leaving 2,000 yen.

Stacked discounts. "30% off then 10% off" - multiply the remaining fractions: 0.7 x 0.9 = 0.63. That is 37% off total.

True discount rate of point rewards. For a reward rate of R%, the effective discount is R / (100 + R) x 100. At 10% reward: 10 / 110 x 100 is roughly 9.1%. Point rewards always deliver a lower effective discount than their face value suggests.

Once you can run these calculations instantly, you can calmly judge whether a sale or coupon is genuinely worth it. Consumers who are comfortable with numbers do not get swept up by marketing.