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Worksheet Grade 7 Math

Ironvale's Sludge Thickener: A Grade 7 Percent-Solids Worksheet

Students work through percent solids, percent decrease, and percent increase using daily sludge and digester gas records from a fictional recovery facility.

Objectives

  1. Students will calculate the pounds of dry solids in a sludge stream from a stated percent solids.
  2. Students will find the percent decrease in sludge weight after thickening.
  3. Students will find the percent increase in daily digester gas between two years.

Vocabulary

Key words: thickener, percent solids, digester, biogas

Teacher overview

The town of Ironvale is fictional, and every number on this worksheet was written for it. Ironvale's water resource recovery facility pumps thin sludge into a thickener, which drains water off, and then feeds the thickened sludge into a digester that produces gas. One misconception is worth planning for. Students see the sludge weight fall from 40,000 pounds to 20,000 pounds and conclude that half the solids were removed. Thickening removes water and leaves the dry solids alone, so the pounds of solids stay constant while the percent solids doubles.

Ironvale's Sludge Thickener: A Grade 7 Percent-Solids Worksheet — Grade 7 Math
Name: ____________________________ Date: ______________

Student Worksheet

Ironvale is a fictional town, and all of these numbers were made up for this worksheet. The Ironvale plant thickens sludge and then digests it. Each item gives you the numbers you need. Round percents as the item tells you.

  1. The thickener takes in 40,000 pounds of sludge each day. That sludge is 2.5 percent solids. How many pounds of dry solids is that?
  2. Use your answer to item 1. Of the 40,000 pounds, how many pounds are water?
  3. After thickening, the sludge weighs 20,000 pounds. It still holds 1,000 pounds of dry solids. What percent solids is the thickened sludge?
  4. The daily weight fell from 40,000 pounds to 20,000 pounds. What is the percent decrease?
  5. The digester is fed 20,000 pounds of thickened sludge each day. The digester holds 300,000 pounds. How many days does a load of sludge stay in the digester?
  6. Next year Ironvale expects 15 percent more dry solids than the 1,000 pounds in item 1. How many pounds of dry solids will arrive each day?
  7. The digester turns 60 percent of the dry solids into gas. Use the 1,150 pounds from item 6. How many pounds of gas is that each day?
  8. Last year the digester turned 55 percent of 1,000 pounds of dry solids into gas. How many pounds of gas was that each day?
  9. Compare items 7 and 8. What is the percent increase in daily gas, from 550 pounds to 690 pounds? Round to the nearest tenth of a percent.
  10. A student says the thickener removes half the solids, because the weight drops from 40,000 pounds to 20,000 pounds. Use items 1 and 3 to explain why that is wrong.

From Water for Schools (waterforschools.org). This material is in the public domain (CC0) and may be copied, edited, and shared freely.

Answer Key
  1. Answer: 1,000 pounds.
    Multiply 40,000 by 0.025. The product is 1,000 pounds of dry solids.
  2. Answer: 39,000 pounds.
    Subtract the dry solids from the total: 40,000 - 1,000 = 39,000 pounds of water.
  3. Answer: 5 percent.
    Divide the solids by the total weight: 1,000 / 20,000 = 0.05. Multiply by 100 to get 5 percent.
  4. Answer: 50 percent.
    The drop is 40,000 - 20,000 = 20,000 pounds. Divide the drop by the original weight: 20,000 / 40,000 = 0.50, which is 50 percent.
  5. Answer: 15 days.
    Divide the tank contents by the daily feed: 300,000 / 20,000 = 15 days.
  6. Answer: 1,150 pounds.
    Fifteen percent of 1,000 pounds is 0.15 x 1,000 = 150 pounds. Add that to the original amount: 1,000 + 150 = 1,150 pounds.
  7. Answer: 690 pounds.
    Multiply the dry solids by the gas fraction: 1,150 x 0.60 = 690 pounds.
  8. Answer: 550 pounds.
    Multiply last year's dry solids by last year's fraction: 1,000 x 0.55 = 550 pounds.
  9. Answer: 25.5 percent.
    The increase is 690 - 550 = 140 pounds. Divide by the starting amount: 140 / 550 = 0.2545. Multiply by 100 to get about 25.5 percent. Students who divide by 690 instead of 550 will get about 20.3 percent, which is the most common error here.
  10. Answer: The thickener removes water, not solids.
    Item 1 shows 1,000 pounds of dry solids entering, and item 3 shows the same 1,000 pounds still present after thickening. The weight fell because 20,000 pounds of water drained away. Since the solids stayed the same while the total shrank, the percent solids doubled from 2.5 percent to 5 percent.

Differentiation

Support

Give students a two-row table for the fictional Ironvale plant with the columns already labeled "pounds of solids," "pounds of water," and "total pounds," and fill in the first row with them before item 1. Students who struggle with percent as a decimal can be given the conversion 2.5 percent equals 0.025 written on the page.

Extension

Ask students to find what percent solids the thickened sludge would reach in this fictional plant if the thickener drained the daily total down to 12,500 pounds while the 1,000 pounds of dry solids stayed the same, then to explain why there is an upper limit on how far a thickener can go.

ELL supports

Preteach "thickener," "digester," and "percent solids" with a labeled flow sketch showing sludge entering the thickener at the town of Ironvale, water draining off, and thickened sludge moving to the digester. Provide a sentence frame for item 10: "The weight dropped because ____, but the pounds of solids stayed at ____."

Extensions

Have students research how a nearby wastewater utility uses digester gas, then return to the fictional Ironvale numbers and estimate how many days of gas production would be needed to match a stated energy use. Students can also rewrite items 6 through 9 for a plant that grows 25 percent instead of 15 percent, and compare how the percent increase in gas changes.