SAT-52: Percent Change — Increase, Decrease, and Successive Changes

SAT-52: Percent Change — Increase, Decrease, and Successive Changes

Description: In SAT-51 you found a part of a whole. This lesson is about how much a quantity changed — a percent increase or percent decrease — and what happens when several percent changes are applied one after another. This is one of the most tested ideas in the whole "Problem Solving and Data" section, so it is worth mastering completely.

The percent change formula

percent change = (new − old) / old × 100%

The denominator is always the original amount (the "old" value), never the new one. If the result is positive it is an increase; if negative it is a decrease.

(Oʻzbekcha: foiz oʻzgarishi har doim boshlangʻich (eski) qiymatga boʻlinadi, yangisiga emas.)

The multiplier shortcut (faster and safer)

Instead of finding the change and adding it back, multiply the original by a single number called the multiplier:

• Increase by r% → multiply by (1 + r/100). Example: +20% → ×1.20.
• Decrease by r% → multiply by (1 − r/100). Example: −15% → ×0.85.

This is the same idea as the growth factor in SAT-45, and it makes successive changes easy. (Oʻzbekcha: oshirish uchun 1 + r/100 ga, kamaytirish uchun 1 − r/100 ga koʻpaytiramiz.)

Worked Example 1 — basic increase

A price rises from $40 to $50. What is the percent increase?

• change = (50 − 40)/40 × 100% = 10/40 × 100% = 25%.

Worked Example 2 — finding the new value with a multiplier

A $120 jacket is discounted 30%. What is the sale price? Decrease 30% → multiplier 0.70. 120 × 0.70 = $84. (Faster than finding $36 off, then subtracting.) Worked Example 3 — successive percent changes (the tricky one) A stock goes up 10%, then down 10%. Is it back to the start? No! Multiply the multipliers: 1.10 × 0.90 = 0.99. So the final value is 99% of the original — a net 1% loss. Key idea: you multiply successive percent changes; you never just add or subtract the percents. (Oʻzbekcha: ketma-ket foizlarni qoʻshmaymiz — koʻpaytuvchilarni koʻpaytiramiz.) Common mistake: thinking +10% then −10% cancels out. Each percent is taken of a different base, so they don't. Choosing the right base The hardest part of percent-change problems is deciding what counts as the "old" value. The base is whatever the change is measured from. "An increase from 40 to 50" makes 40 the base. "50 is what percent more than 40?" also makes 40 the base. But "40 is what percent less than 50?" switches the base to 50 — so the answer changes. Always underline the word right after "than" or "from"; that is your denominator. This single habit prevents most percent-change errors on the test. (Oʻzbekcha: "than" yoki "from" soʻzidan keyingi son — bu maxraj (asos) boʻladi.) Practice 1 A population falls from 800 to 600. What is the percent decrease? Show answer (600 − 800)/800 × 100% = −200/800 × 100% = −25%, i.e. a 25% decrease. Practice 2 A $200 item is increased 25%, then that price is decreased 20%. Final price?

Key words — Kalit soʻzlar

• Percent change — foiz oʻzgarishi
• Increase — oshish
• Decrease — kamayish
• Original (old) value — boshlangʻich (eski) qiymat
• Multiplier — koʻpaytuvchi koeffitsiyent
• Discount — chegirma
• Successive — ketma-ket
• Net change — yakuniy (sof) oʻzgarish
• Sale price — chegirmali narx

Summary

• Percent change = (new − old)/old × 100% — always divide by the original.
• Use multipliers: +r% → ×(1 + r/100); −r% → ×(1 − r/100).
• For successive changes, multiply the multipliers; never add the percents.