Problem 1: Percent increase
Change: \(5-4=1\). Divide by the original: \(\frac{1}{4}=0.25=25\%\).
The price rose, so a 25% increase.
Problem 2: Gas price drop
Change: \(4.00-3.60=0.40\) dollars. Divide by the original price: \(\frac{0.40}{4.00}=0.10=10\%\).
A 10% decrease.
Problem 3: Percent decrease
Change: \(50-40=10\) gallons. Divide by the original: \(\frac{10}{50}=0.2=20\%\).
A 20% decrease.
Problem 4: Growth of a seedling
Change: \(28-25=3\) inches. Divide by the original height: \(\frac{3}{25}=0.12=12\%\).
A 12% increase.
Problem 5: Falling attendance
Change: \(250-215=35\) fans. Divide by the original attendance: \(\frac{35}{250}=0.14=14\%\).
A 14% decrease.
Problem 6: Price after a raise
Increase: \(0.15\times 60=9\). New price: \(60+9=69\). Check with the multiplier: \(1.15\times 60=69\).
The new price is $69.
Problem 7: New monthly fee
Increase: \(0.25\times 14=3.50\). New fee: \(14+3.50=17.50\). Check with the multiplier: \(1.25\times 14=17.50\).
The new fee is $17.50.
Problem 8: Time after a decrease
Decrease: \(0.30\times 90=27\) minutes. New time: \(90-27=63\) minutes. Check: \(0.70\times 90=63\).
63 minutes.
Problem 9: Sale price
Discount: \(0.20\times 85=17\). Sale price: \(85-17=68\).
The sale price is $68.
Problem 10: Estimate error
Error: \(|450-500|=50\) people. Divide by the actual value: \(\frac{50}{500}=0.10=10\%\).
The percent error is 10%.
Problem 11: Percent error
Error: \(|12.6-12|=0.6\). Divide by the actual value: \(\frac{0.6}{12}=0.05=5\%\).
The percent error is 5%.
Problem 12: Find the original
The new price is \(120\%\) of the original, so \(1.20\times \text{original}=60\).
\(\text{original}=60: 1.20=50\).
Check: \(20\%\) of $50 is $10 and \(50+10=60\). The original price was $50.
Problem 13: Price before the sale
The sale price is \(100\%-15\%=85\%\) of the regular price, so \(0.85\times \text{regular price}=68\).
\(\text{regular price}=68: 0.85=80\).
Check: \(15\%\) of $80 is $12 and \(80-12=68\). The regular price was $80.
Problem 14: Which rose more?
- Theater A: \(\frac{36-30}{30}=\frac{6}{30}=0.20=20\%\). Theater B: \(\frac{58-50}{50}=\frac{8}{50}=0.16=16\%\).
- Theater A has the greater percent increase. Theater B rose by more dollars ($8 compared with $6), but its original price is much larger, so each dollar is a smaller part of it.
(a) 20% and 16% (b) Theater A.
Problem 15: Population change
- Change: \(9{,}200-8{,}000=1{,}200\). \(\frac{1{,}200}{8{,}000}=0.15=15\%\). A 15% increase.
- Decrease: \(0.10\times 9{,}200=920\). \(9{,}200-920=8{,}280\). 8,280 residents.
Problem 16: Discount and tax
- \(0.75\times \text{original}=45\), so \(\text{original}=45: 0.75=60\). Check: \(25\%\) of $60 is $15 and \(60-15=45\).
- Tax: \(0.08\times 45=3.60\). Total: \(45+3.60=48.60\), or \(1.08\times 45=48.60\).
(a) $60 (b) $48.60.
Problem 17: Up then down
- Rise: \(0.50\times 40=20\). Value: \(40+20=60\).
- Fall: \(0.50\times 60=30\). Value: \(60-30=30\).
- Overall change: \(30-40=-10\). \(\frac{-10}{40}=-0.25\), a \(25\%\) decrease.
(a) $60 (b) $30 (c) 25% decrease overall.
Problem 18: Markup and discount
- Markup: \(0.75\times 24=18\). Marked price: \(24+18=42\).
- Discount: \(0.20\times 42=8.40\). Sale price: \(42-8.40=33.60\).
- Change from the cost: \(33.60-24=9.60\). \(\frac{9.60}{24}=0.40=40\%\).
(a) $42 (b) $33.60 (c) 40% greater.
Problem 19: Two years of growth
- Year 1: \(1.20\times 5{,}000=6{,}000\). Year 2: \(1.10\times 6{,}000=6{,}600\).
- Change: \(6{,}600-5{,}000=1{,}600\). \(\frac{1{,}600}{5{,}000}=0.32=32\%\).
- The second \(10\%\) is taken of 6,000, not of 5,000, so it adds 600 instead of 500. The increases do not simply add: \(1.20\times 1.10=1.32\).
(a) 6,000 and 6,600 (b) 32% (c) the percents apply to different amounts.
Problem 20: Bookstore markup
- Markup: \(14-8=6\). \(\frac{6}{8}=0.75=75\%\).
- Discount: \(0.30\times 14=4.20\). Reduced price: \(14-4.20=9.80\).
- Change from the cost: \(9.80-8=1.80\). \(\frac{1.80}{8}=0.225=22.5\%\).
(a) 75% (b) $9.80 (c) 22.5% greater.
See every chapter on the Grade 7 math page.